Finite Element Analysis and Active Control for Nonlinear

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Finite Element Analysis and Active Control forNonlinear Flutter of Composite Panels UnderYawed Supersonic FlowKhaled Abdel-MotagalyOld Dominion University

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Recommended CitationAbdel-Motagaly, Khaled. "Finite Element Analysis and Active Control for Nonlinear Flutter of Composite Panels Under YawedSupersonic Flow" (2001). Doctor of Philosophy (PhD), dissertation, Mechanical & Aerospace Engineering, Old DominionUniversity, DOI: 10.25777/1fvd-ra53https://digitalcommons.odu.edu/mae_etds/210

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FINITE ELEMENT ANALYSIS AND ACTIVE CONTROL

FOR NONLINEAR FLUTTER OF COMPOSITE PANELS

UNDER YAWED SUPERSONIC FLOW

Khaled Abdel-MotagalyB.S. July 1991, Cairo University, Egypt M.S. July 1995, Cairo University, Egypt

A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the

Requirement for the Degree of

DOCTOR OF PHILOSOPHY

AEROSPACE ENGINEERING

OLD DOMINION UNIVERSITY December 2001

by

Approved by :

Chuh Mei (Director)

Osama Kandil (Member)

Jen-K.uang Huang (Member)

Donald Kunz (Member^

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

ABSTRACT

FINITE ELEMENT ANALYSIS AND ACTIVE CONTROL FOR NONLINEAR FLUTTER OF COMPOSITE PANELS

UNDER YAWED SUPERSONIC FLOW

Khaled Abdel-Motagaly Old Dominion University, 2001

Director: Dr. Chuh Mei

A coupled structural-electrical modal finite element formulation for composite

panels with integrated piezoelectric sensors and actuators is presented for nonlinear panel

flutter suppression under yawed supersonic flow. The first-order shear deformation

theory for laminated composite plates, the von Karman nonlinear strain-displacement

relations for large deflection response, the linear piezoelectricity constitutive relations,

and the first-order piston theory of aerodynamics are employed. Nonlinear equations of

motion are derived using the three-node triangular MIN3 plate element. Additional

electrical degrees o f freedom are introduced to model piezoelectric sensors and actuators.

The system equations o f motion are transformed and reduced to a set of nonlinear

equations in modal coordinates. Modal participation is defined and used to determine the

number of modes required for accurate solution.

Analysis results for the effect of arbitrary flow yaw angle on nonlinear supersonic

panel flutter for isotropic and composite panels are presented. The results show that the

flow yaw angle has a major effect on the panel limit-cycle oscillation amplitude and

deflection shape. The effect of combined aerodynamic and acoustic pressure loading on

the nonlinear dynamic response of isotropic and composite panels is also presented. It is

found that combined acoustic and aerodynamic loads have to be considered for high

aerodynamic pressure values.

Simulation studies for nonlinear panel flutter suppression using piezoelectric self

sensing actuators under yawed supersonic flow are presented for isotropic and composite

panels. Different control strategies are considered including linear quadratic Gaussian

(LQG), linear quadratic regulator (LQR) combined with the extended Kalman filter

(EKF), and optimal output feedback. Closed loop criteria based on the norm of feedback


control gain (NFCG) and on the norm o f Kalman filter estimator gain (NKFEG) are used

to determine the optimal location of piezoelectric actuators and sensors, respectively.

Optimal sensor and actuator locations for a range of yaw angles are determined by

grouping the optimal locations for different angles within the range. The results

demonstrate the effectiveness o f piezoelectric materials and of the nonlinear output

controller comprised o f LQR state feedback and EKF nonlinear state estimator in

suppressing nonlinear flutter o f isotropic and composite panels at different flow yaw

angles.


ACKNOWLEDGMENTS

I would like to express my sincere thanks and deepest appreciation to Professor

Chuh Mei for his everlasting assistance, encouragement, support, and patience. His

guidance, inspiring ideas, and enlightening discussion played a key role in the production

of this work. I would also like to extend my sincere thanks and deep gratitude to the other

members o f my dissertation committee: Professors Osama Kandil, Donald Kunz, and

Jen-Kuang Huang. Their invaluable comments and advice are much appreciated and are

an added value to this dissertation. I gratefully acknowledge the financial support of the

Aerospace Engineering Department during the course of my research.

I extend my thanks and gratitude to Professor Sayed Dessoki, who taught me a lot

during my entire career. Last, but not least, special acknowledgment and thanks go to my

parents and my wife for their everlasting love and encouragement. Without their support,

this work would have been impossible.


TABLE OF CONTENTS

Page

LIST OF TA B LE S.......................................................................................................................viii

LIST OF FIG U RES....................................................................................................................... ix

LIST OF SY M BO LS....................................................................................................................xiv

Chapter

I. INTRODUCTION AND LITERATURE SU R V EY ...................................................... 1

1.1 Introduction.................................................................................................................. 1

1.2 Background and Literature S u rv ey ........................................................................... 2

1.2.1 Panel F lu tte r .....................................................................................................2

1.2.2 Piezoelectric Sensors and A ctuators.............................................................8

1.2.3 Panel Flutter Suppression Using Piezoelectric M aterials..................... 11

1.3 Outline of the S tu d y ................................................................................................ 13

II. FINITE ELEMENT FO RM U LATIO N........................................................................ 19

2.1 Introduction..................................................................................................................19

2.2 Element Displacement Functions........................................................................... 20

2.3 Nonlinear Strain-Displacement R elations............................................................ 22

2.4 Electrical Field-Potential R elations........................................................................23

2.5 Linear Piezoelectricity Coupling E quations.........................................................24

2.6 Constitutive E quations..............................................................................................25

2.7 Laminate Resultant Forces and M om ents.............................................................26

2.8 Aerodynamic Pressure L oading ..............................................................................27

2.9 Element Equations o f Motion and M atrices.........................................................28

2.9.1 Generalized Hamilton’s P rincip le ............................................................ 28

2.9.2 Element Stiffness and Electromechanical Coupling M atrices...........29

2.9.3 Element Mass M atrices..............................................................................34

2.9.4 Element Aerodynamic M atrices .............................................................. 35

2.9.5 Element Equations of M o tio n .................................................................. 36

2.10 System Equations of M otion ...................................................................................37

2.11 Piezoelectric Actuator and Sensor E quations......................................................38


2.11.1 Piezoelectric Material as Actuators and S enso rs .................................. 38

2.11.2 Piezoelectric Material as Self-Sensing A ctuators.................................40

III. MODAL REDUCTION AND SOLUTION PRO CED U RE.......................................45

3.1 Introduction..................................................................................................................45

3.2 Governing Equations................................................................................................. 46

3.3 Modal Transformation and R eduction................................................................... 48

3.4 Solution Procedure..................................................................................................... 51

3.4.1 Time Domain M eth o d ................................................................................51

3.4.2 Frequency Domain M ethod .......................................................................51

3.4.2.1 Critical Flutter B oundary.......................................................... 51

3.4.2.2 Nonlinear Flutter Limit-Cycle O scilla tion ............................ 53

IV. FINITE ELEMENT ANALYSIS R ESU LTS................................................................57

4.1 Introduction................................................................................................................. 57

4.2 Finite Element V alidation.........................................................................................57

4.2.1 Natural Frequencies....................................................................................57

4.2.2 Piezoelectric Static A ctuation .................................................................. 58

4.2.3 Linear and Nonlinear F lu tte r.................................................................... 58

4.2.4 Nonlinear Random R esponse................................................................... 59

4.3 Effect o f Flow Yaw angle on Nonlinear Panel F lu tte r........................................ 60

4.3.1 Isotropic P anels........................................................................................... 60

4.3.2 Composite P an e ls ....................................................................................... 61

4.3.3 Triangular P an e l..........................................................................................62

4.4 Effect of Combined Supersonic Flow and Acoustic Pressure L oad ing 62

4.4.1 Random Surface P ressu re .........................................................................62

4.4.2 Isotropic P an e l.............................................................................................63

4.4.3 Composite P an e l.........................................................................................64

4.5 Sum m ary..................................................................................................................... 65

V. CONTROL METHODS AND OPTIMAL PLACEMENT

OF PIEZOELECTRIC SENSORS AND A C TA U TO RS........................................... 99

5.1 Introduction................................................................................................................ 99

5.2 State Space Representation..................................................................................... 99


5.3 Control M ethods................................................................................................... 101

5.3.1 Linear Quadratic Regulator (L Q R )...................................................... 101

5.3.2 Optimal Linear State Estim ation.......................................................... 102

5.3.3 Linear Quadratic Gaussian Controller (L Q G )................................... 103

5.3.4 Extended Kalman Filter (E K F )............................................................ 104

5.3.5 Nonlinear Controller using EKF and L Q R ........................................ 104

5.3.6 Optimal Output Feedback..................................................................... 105

5.4 Optimal Placement of Piezoelectric Sensors and A ctuators......................... 105

VI. NONLINEAR PANEL FLUTTER SUPPRESSION R E SU LT S.......................... 109

6.1 Introduction............................................................................................................ 109

6.2 Square Isotropic P anel......................................................................................... 109

6.2.1 Optimal Placement of Self-Sensing Piezoelectric A ctuators......... 110

6.2.2 Controller S tudy ...................................................................................... I l l

6.2.2.1 LQR C ontroller....................................................................... 112

6.2.2.2 LQG C ontroller....................................................................... 113

6.2.2.3 Nonlinear Output Controller (EK F+LQ R )......................... 113

6.2.2.4 Optimal Output Feedback C ontro ller.................................. 114

6.2.2.5 Controller Robustness............................................................ 115

6.2.3 Panel Flutter Control with Yawed F lo w ............................................ 116

6.3 Composite P an e l................................................................................................. 118

6.4 Triangular P an e l................................................................................................. 119

6.5 S um m ary............................................................................................................. 120

VII. SUMMARY AND CONCLUSIONS......................................................................... 158

R EFER EN C ES........................................................................................................................... 161

APPENDCIES ............................................................................................................................ 172

A. MIN3 ELEMENT STRAIN INTERPOLATION M A TR IC ES............................ 172

B. COORDINATE TRANSFORM ATION.................................................................... 174

B .l Transformation of Lamina Stiffness M atrices................................................. 174

B.2 Transformation of Piezoelectric C onstants....................................................... 175

C. SOLUTION PROCEDURE FOR OPTIMAL OUTPUT FEED BA CK .............. 176

V IT A ............................................................................................................................................. 177


LIST OF TABLES

Table Page

1.1 Panel Flutter T heories ................................................................................................... 16

1.2 Analogy between electrical and mechanical variables

for piezoelectric m ateria ls............................................................................................. 16

4.1 Material properties for the different materials used for

Finite element validation and analysis resu lts ............................................................. 66

4.2 Comparison between natural frequencies using MIN3 element

and using analytical solution for square isotropic p an e l........................................... 67

4.3 Comparison of static deflection for the piezoelectric bimorph

beam (10'7 m) using different m ethods........................................................................ 67

4.4 Comparison of nondimensional linear flutter boundary using

different methods at different flow an g les ...................................................................68

4.5 Comparison of the RMS (Wmax/h) for a simply supported rectangular

(15x12x0.04 in.) isotropic plate under acoustic pressure loading only

using different methods and mode num bers................................................................68

4.6 Modal participation values at various limit-cycle amplitudes and different

flow angles for simply supported square p an el........................................................... 69

4.7 Modal participation values at various limit-cycle amplitudes and different flow

angles for simply supported rectangular graphite/epoxy [0/45/-45/90]s panel ... 70

4.8 Modal participation values at various limit-cycle amplitudes and different flow

angles for simply supported rectangular graphite/epoxy [-40/40/-40] p an e l 71

4.9 Modal convergence for clamped rectangular [0/45/-45/90]s graphite/epoxy

panel at SPL = 120 dB and X = 800 ............................................................................. 72

4.10 Modal participation for a clamped rectangular [0/45/-45/90]s graphite/epoxy

panel at SPL = 120 dB and X = 800 .............................................................................. 72

6.1 Mechanical and electrical properties for PZT5A piezoelectric ceram ics 122

6.2 Comparison of different controllers performance for nonlinear

panel flutter suppression.............................................................................................. 122


LIST OF FIGURES

Figure Page

1.1 Explanation of nonlinear panel flutter phenom enon................................................. 17

1.2 Schematic of traditional isotropic piezoelectric elem ent.......................................... 18

2.1 Composite panel under yawed supersonic flow and acoustic

pressure loading ................................................................................................................ 42

2.2 Composite laminate composed of n layers with n p piezoelectric lay e rs .................43

2.3 MIN3 element geometry and area coordinates............................................................44

3.1 Configuration of self-sensing piezoelectric actuators for bending moment

actuation and transverse displacement sen sin g ..........................................................55

3.2 Iterative solution procedure for nonlinear panel flutter limit-cycle resp o n se 56

4.1 Typical MIN3 elements mesh used to model rectangular and

triangular panels.................................................................................................................73

4.2 Clamped piezoelectric bimorph beam modeled using MIN3 elem ents..................74

4.3 Validation of flutter limit-cycle amplitude for simply supported square

isotropic p a n e l.................................................................................................................... 75

4.4 Validation of flutter limit-cycle amplitude for simply supported square

[0/45/-45/90]s lam inate.....................................................................................................76

4.5 Finite element mesh convergence for simply supported isotropic square

panel at 45° flow angle and using 16 m o d es ................................................................ 77

4.6 Modal convergence for simply supported isotropic

square panel at 0° flow a n g le .......................................................................................... 78

4.7 Modal convergence for simply supported isotropic

square panel at 45° flow a n g le ........................................................................................ 79

4.8 Effect of flow yaw angle on limit-cycle amplitude for

simply supported isotropic square p a n e l....................................................................... 80

4.9 Effect o f panel aspect ratio at different yaw angles on limit-cycle

amplitude for simply supported isotropic p a n e ls ........................................................81

4.10 Flutter mode shape at different flow angles for simply supported

isotropic square p an e l.......................................................................................................82


X

4 .1 1 Modal convergence for simply supported rectangular graphite/epoxy

[0/45/-45/90]s panel at 45° flow a n g le .......................................................................... 83

4.12 Effect of flow yaw angle on limit-cycle amplitude for simply

supported rectangular graphite/epoxy [0/45/-45/90]s p a n e l.....................................84

4.13 Flutter mode shape at different flow angles for simply supported

rectangular graphite/epoxy [0/45/-45/90]s p a n e l........................................................ 85

4.14 Modal convergence for simply supported rectangular graphite/epoxy

[-40/40/-40] panel at 45° flow a n g le ..............................................................................86

4.15 Effect of flow yaw angle on limit-cycle amplitude for simply

supported rectangular graphite/epoxy [-40/40/-40] p a n e l......................................... 87

4.16 Flutter mode shape at different flow angles for simply supported rectangular

graphite/epoxy [40/-40/40] p an e l....................................................................................88

4.17 Modal convergence for simply supported triangular isotropic panel

at 45° flow a n g le ................................................................................................................89

4.18 Effect o f flow yaw angle on limit-cycle amplitude for simply supported

triangular isotropic p an e l................................................................................................. 90

4.19 Power spectral density of random input pressure at SPL = 9 0 d B .......................... 91

4.20 RMS of maximum deflection for simply supported square isotropic panel

under combined acoustic and aerodynamic pressu res...............................................92

4.21 Time and frequency response for simply supported square isotropic panel

at SPL = 100 dB and X = 0 ...............................................................................................93

4.22 Time and frequency response for simply supported square

isotropic panel at SPL = 120 dB and X = 0 .................................................................. 94


isotropic panel at SPL = 0 dB and X = 800 ......................................................... 95


isotropic panel at SPL = 100 dB and X = 800 .................................................... 96


isotropic panel at SPL = 120 dB and X = 800 .................................................... 97

4.26 RMS of maximum deflection for clamped rectangular

graphite/epoxy [0/45/-45/90]s p a n e l.............................................................................98


5.1 Block diagram representation of the LQG contro ller............................................ 107

5.2 Nonlinear dynamic compensator for nonlinear panel flutter

suppression using EKF and L Q R ............................................................................... 108

6.1 Contours o f (a) NFCG and (b) NKFEG for simply supported square

isotropic panel at 0° flow an g le .................................................................................. 123

6.2 Contours o f (a) NFCG and (b) NKFEG for simply supported square

isotropic panel at 45° flow an g le ................................................................................ 124

6.3 Selected self-sensing piezoelectric actuators placement and size for optimal

actuation and optimal sensing on square isotropic panel at 0° flow an g le 125

6.4 Self-sensing piezoelectric actuators placement and size for optimal

actuation and optimal sensing on square isotropic panel at 45° flow an g le 126

6.5 Variation of first 4 linear modes versus X for isotropic panel with

and without added piezoelectric m aterial................................................................ 127

6.6 Open loop poles for isotropic square panels with embedded piezoelectric

material at different values of nondimensional dynamic p ressu re ....................... 128

6.7 Limit-cycle amplitude and control inputs time history for square isotropic

panel using LQR control at X = 1500 ........................................................................ 129

6.8 Comparison between actual LCO amplitude and estimated LCO amplitude

using Kalman filter at X = 1000 ................................................................................. 130

6.9 Comparison between actual LCO amplitude and estimated LCO amplitude

using Kalman filter at X = 2000 ................................................................................. 131

6.10 Performance of LQG controller at X = 1500 ........................................................... 132

6 .11 Performance of LQG controller at the maximum suppressible

dynamic pressure, Amax = 920 .................................................................................. 133

6.12 Comparison between actual and estimated LCO amplitude

using extended Kalman filter at X = 2000 ................................................................ 134

6.13 Performance of LQR+EKF nonlinear output compensator at X = 1500 ............ 135

6.14 Performance of LQR+EKF nonlinear output controller at X = 1500 with

(a) -25% and (b) +25% uncertainty in the model nonlinear stiffness matrix .... 136

6.15 Performance of optimal output feedback controller using two

self-sensing actuators at Amax = 1000 ...................................................................... 137


6.16 Performance of optimal output feedback controller using single leading edge

actuator and two displacement sensors at Amax = 11 0 0 ........................................ 138

6.17 Effect of +25% mismatch in A between design model and simulation model

on LQR and LQR+EKF control performance at A = 1200 .................................... 139

6.18 Effect of design model linear stiffness variation on The LQR+EKF

controller performance at A = 1 5 0 0 ........................................................................... 140

6.19 Effect of design model linear stiffness variation on optimal output

feedback controller performance at A = 1100.......................................................... 141

6.20 Performance of EKF+LQR controller designed for zero flow angle

at A = 800 and 45 deg flow a n g le ................................................................................ 142

6.21 Optimal actuator and sensor placement at different flow angles

from 0 to 90° for square isotropic p a n e l..................................................................... 143

6.22 Placement of four self-sensing piezoelectric actuators for optimal

actuation and optimal sensing over the range of [0, 90°] flow

angle for square isotropic p a n e l.................................................................................. 144

6.23 Comparison of panel flutter suppression performance using LQR+EKF

control at different flow yaw angles and using different piezoelectric

placement configurations for a square isotropic p a n e l.......................................... 145

6.24 Performance of LQR+EKF controller for square isotropic panel with 4

self-sensing piezoelectric actuators at 0° flow yaw angle and A = 1500 ........... 146


self-sensing piezoelectric actuators at 45° flow yaw angle and A = 1500 ......... 147


self-sensing piezoelectric actuators at 90° flow yaw angle and A = 1500 ......... 148

6.27 Optimal actuator placement at different flow angles for [0/45/-45/90]s

composite rectangular p a n e l ........................................................................................ 149

6.28 Optimal placement of 2 embedded piezoelectric actuators that cover flow

angles from 0° to 90° for [0/45/-45/90]s composite rectangular p a n e l................150

6.29 Comparison of panel flutter suppression performance using LQR+EKF

control at different flow yaw angles and using different piezoelectric

actuator configurations for [0/45/-45/90]s rectangular composite p a n e l 151


Xlll

6.30 Performance o f LQR+EKF controller for rectangular composite panel

at 0° flow yaw angle and X = 900 ............................................................................... 152

6.31 Performance o f LQR+EKF controller for rectangular composite panel

at 45° flow yaw angle and X = 900 ............................................................................ 153

6.32 Performance of LQR+EKF controller for rectangular composite panel

at 90° flow yaw angle and X = 900 ............................................................................ 154

6.33 Optimal actuator placement at different flow angles

for clamped triangular isotropic p a n e l....................................................................... 155

6.34 Optimal placement o f a single self-sensing piezoelectric actuator that

approximately cover all angles from 90° to 180° for

triangular clamped isotropic p an e l.............................................................................. 156

6.35 Performance of the LQR+EKF controller in suppressing nonlinear

panel flutter using different piezoelectric actuator configurations

for the clamped triangular isotropic p a n e l................................................................ 157


1

CHAPTER I

INTRODUCTION AND LITERATURE SURVEY

1.1 Introduction

Recently, there has been a renewed interest in flight vehicles that operate at high

supersonic and hypersonic Mach numbers, such as the X-38 Crew Return Vehicle

spacecraft for the International Space Station, the X-33 Advanced Technology

Demonstrator, the X-34 Reusable Technology Demonstrator for a launch vehicle, and the

recent NASA Space Launch Initiative (SLI) project. The exterior panels of such vehicles

will be affected by supersonic panel flutter phenomena. These flight vehicles will usually

operate for a range of flow yaw angles and will also be subjected to additional loading

due to random pressure fluctuations (sonic fatigue). This brings an urgent need for panel

flutter analysis at supersonic speeds considering the effect of flow yaw angle and the

effect of additional acoustic loading.

The requirements of energy-efficient, high-strength, and minimum-weight

vehicles have generated an interest in advanced lightweight composite materials. In

addition, higher performance can be obtained by using the recently developed smart or

adaptive materials such as piezoelectric ceramics that are embedded into the laminated

composite panels to control and suppress undesired panel vibrations.

The primary objectives of this study are: (1) to develop a finite element tool for

analyzing nonlinear supersonic flutter of composite panels considering the effects of flow

yaw angle and the effect of additional acoustic loading and (2) to design practical control

methodologies that suppress nonlinear supersonic panel flutter o f composite and isotropic

panels using piezoelectric sensors and actuators considering the effect of flow yaw angle.

The next sections present an overview and literature survey for the main topics of this

research including classical and finite element analysis methods for nonlinear panel

flutter, piezoelectric sensors and actuators, and panel flutter suppression using

piezoelectric materials. An outline of the dissertation is then given at the end of the

chapter.

The journal model used for this dissertation is the AlAA Journal.


1.2 Background and Literature Survey1.2.1 Panel Flutter

Supersonic panel flutter is a self-excited oscillation of panels exposed to

aerodynamic flow with high Mach numbers. Figure 1.1 shows four different schematics

explaining the panel flutter phenomenon. For dynamic pressures, q , less than the flutter

boundary, random pressure fluctuations due to turbulent boundary layer control the panel

response. At this regime, the panel response can be determined using standard linear

sonic fatigue (noise) analysis techniques and is usually in the small displacement region,

i.e., maximum panel displacement divided by panel thickness { W m io / h ) is much less than

one. As the dynamic pressure increases, the panel stiffness is modified by the

aerodynamic loading such that the first mode natural frequency, co, increases while thd*

second mode natural frequency decreases. At the flutter boundary, the two modes

coalesce and the panel motion becomes unstable, based on linear structure theory.

However, due to the structural nonlinearities (inplane stretching forces) and unlike the

catastrophic failure for wing flutter, the panel motion is limited to a constant amplitude

oscillation. The inplane stretching forces tend to restrain the panel motion so that

bounded limit-cycle oscillations (LCO) are observed as shown in Figure 1.1. The

amplitude of the LCO grows as the dynamic pressure increases. The existence of LCO

implies that large deflection nonlinear structural theory should be used beyond the flutter

critical dynamic pressure to estimate panel response and fatigue life. The flutter LCO

deflection shape depends on many factors such as flow yaw angle, panel boundary

conditions, and composite laminate stacking. An example of flutter deflection shape for

an isotropic simply supported square panel is shown in Figure 1.1.

Since the late fifties and early sixties, there have been many articles in the

literature addressing linear and nonlinear panel flutter. An excellent review article for

linear and nonlinear panel flutter theories and analysis through 1970 is given by Dowell

[1]. Recently, Mei et al. [2], have produced an extensive review of various analytical and

experimental results for nonlinear supersonic and hypersonic panel flutter up to 1999.

Dowell [1] has grouped the vast amount of theoretical literature on panel flutter into four

categories based on the structural and aerodynamic theories used. Gray and Mei [3]


3

added a fifth category for hypersonic flow. The five different categories o f linear and

nonlinear panel flutter are shown in Table 1.1. The weakness and remedies for the first

four types o f analysis were discussed in detail by Dowell. A review of the finite element

method of type-1 panel flutter analysis was given by Bismark-Nasr [4], A survey on

various analytical methods, including finite element method for nonlinear supersonic

panel flutter type-3 analysis, was given by Zhou et al. [5]. The fundamental theories and

physical understanding of panel flutter are given in detail in published books, [6] and [7],

This study is concerned with the type-3 panel flutter analysis that uses nonlinear structure

theory and linear piston theory of aerodynamics with yawed supersonic flow.

As disclosed by these survey papers, a vast quantity of literature exists on panel

flutter using different aerodynamic theories. The aerodynamic theory employed for the

most part of panel flutter at high supersonic Mach numbers (M„ > 1.6) is the quasi-steady

first order piston theory developed by Ashley and Zartarian [8], If aerodynamic damping

is neglected, the quasi-steady piston theory simplifies to the quasi-static Ackeret theory.

The piston theory, although several decades old, has generally been employed to

approximate the aerodynamic loads on the panel from local pressures generated by the

body’s motion as related to the local normal component of the fluid velocity and the local

pressure. For supersonic Mach numbers, the quasi-steady aerodynamic theory reasonably

estimates the aerodynamic pressures and shows fair agreement between theory and

experiment for plates exposed to static pressure loads and buckled by uniform thermal

expansion, as was shown by Ventres and Dowell [9]. For airflow with Mach numbers

close to one, the full-linearized inviscid potential theory of aerodynamics is usually

employed [10]. For hypersonic panel flutter, the nonlinear unsteady third-order piston

theory is used to develop the aerodynamic pressure, [11] and [3].

The partial nonlinear behavior of a fluttering panel was first considered by several

investigators such as [12-14], They were primarily concerned with determining stability

boundaries o f two-dimensional plates. For nonlinear limit-cycle behavior, a variety of

methods have been employed to assess the panel flutter problem. Galerkin’s method was

used to reduce the governing partial differential equations to a set of coupled ordinary

differential equations in time, which were numerically integrated using arbitrary initial

conditions. The integration was continued until a limit-cycle oscillation of constant


4

amplitude, independent of the initial conditions, was reached. The nonlinear oscillations

of simply supported [15], and clamped [16, 17] fluttering plates were studied using this

method. Dowell [15] determined that the direct numerical integration approach required a

minimum of 6 linear modes, as the Galerkin approximate functions, to achieve a

converged solution for displacements. Recently, the limit-cycle oscillation of a cantilever

plate was studied by Weiliang and Dowell [18]. They employed a Rayleigh-Ritz

approach in conjunction with the direct numerical integration and showed that the length-

to-width ratio of the cantilever plate was a significant factor on the flutter vibration.

Various techniques in the temporal domain such as harmonic balance and

perturbation techniques have been successfully employed to study the problem of

nonlinear panel flutter. The harmonic balance method requires less computational time

than the method o f direct integration and is mathematically comprehensible and

systematic, but it is extremely tedious to implement. The method was used by Bolotin

[14] and Kobayashi [17] with two-mode Galerkin solution to obtain the limit-cycle

motions. Rectangular plates were treated by Kuo et al. [19], Eastep and McIntosh [20],

Eslami and Ibrahim [21], and Yuen and Lau [22]. The Rayleigh-Ritz approximation to

Hamilton’s variational principle was employed by Eastep and McIntosh to obtain the

equations of motion in the spatial domain. Special orthotropic panels were studied by

Eslami and Ibrahim. A hinged two-dimensional fluttering plate with moderately high

postbuckling loads using a four-mode expansion and an incremental harmonic balance

method was reported by Yuen and Lau. The perturbation method was employed to the

problem of nonlinear panel flutter by Morino [23] and Kuo et al. [19], Detailed

extensions and stability analysis of this technique to nonlinear panel flutter were studied

by Morino and Kuo [24] and Smith and Morino [25]. Correlation between perturbation

techniques and the harmonic balance method has been shown to be in good agreement by

Kuo et al. [19] and Morino and Kuo [24].

All of the early studies in nonlinear panel flutter using classical methods have

been limited to isotropic or orthotropic, two or three-dimensional, rectangular plates with

all four edges simply supported or clamped. Extension of the finite element method to

study the linear panel flutter problem was due to Olson [26, 27] using a frequency

domain eigenvalue solution. Because of its versatile applicability, effects of aerodynamic


5

damping, complex panel configurations and support conditions, laminated composite

anisotropic panel properties, flow angularities, inplane stresses, and thermal loads can be

easily and conveniently included in the finite element formulation. A survey on the finite

element methods for linear panel flutter was given by Yang and Sung [28] and Bismark-

Nasr [4], and for nonlinear panel flutter by Zhou et al. [5]. Application of the finite

element method to study the supersonic limit-cycle oscillations of two-dimensional

panels was given by Mei [29] using an iterative frequency domain solution. Mei and

Rogers [30] implemented the two-dimensional panel flutter analysis into NASTRAN.

Rao and Rao [31] investigated the supersonic flutter of two-dimensional panels with ends

restrained elastically against rotation. Sarma and Varadan [32] studied the nonlinear

behavior of two-dimensional panels using two solution procedures, both in the frequency

domain. Further extension of the finite element method to treat supersonic limit-cycle

oscillations o f three-dimensional rectangular plates was given by Mei and Weidman [33],

The effects of damping, aspect ratio, inplane forces, and boundary conditions were

considered. Mei and Wang [34] employed an 18-degree of freedom (DOF) triangular

plate bending element to study supersonic limit-cycle behavior of three-dimensional

triangular plates. Han and Yang [35] used the 54-DOF high order triangular plate element

to study nonlinear panel flutter of three-dimensional rectangular plates with inplane

forces.

Few papers in the literature have investigated supersonic limit-cycle oscillations

o f composite panels. Dixon and Mei [36] studied the nonlinear flutter of rectangular

composite panels. The limit-cycle response was obtained using a 24-DOF rectangular

plate element and a linearized updated mode with nonlinear time function (LUM/NTF)

approximate solution procedure. The LUM/NTF solution procedure in the frequency

domain was developed by Gray [11]. Because of the renewed interest in panel flutter at

high-supersonic/hypersonic speeds [37], Gray et al. [3] and [11] extended the finite

element method to investigate the hypersonic limit-cycle oscillations of composite panels

using the full third-order piston aerodynamic theory. In practice, aerodynamic heating

will cause thermal loading on the panel in addition to the aerodynamic loading. Xue et al.

[38, 39] investigated flutter boundaries of thermally buckled two-dimensional and three-

dimensional isotropic panels of arbitrary shape using the discrete Kirchhoff theory (DKT)


6

triangular plate element. The finite element equations in structure node DOF were

separated into two sets of equations and then solved sequentially. The first set of

equations yields the thermal-aerodynamic equilibrium position using Newton-Raphson

iterative method, and the second set of equations leads to the flutter limit-cycle motions

using the LUM/NTF approximate method. The use of LUM/NTF approximate method

has been successful in studying nonlinear panel flutter. However, the application of the

LUM/NTF method to the system equations has three disadvantages: (1) the number of

structure node DOF of {W } is usually very large, (2) at each iteration, the element

nonlinear stiffness matrices have to be evaluated and the system nonlinear matrices have

to be assembled, and (3) the periodic and chaotic panel motions can not be determined.

Zhou et al. [40] introduced a solution to these problems by transforming the structure

DOF system equations o f motion into a set of modal coordinates of rather small DOF.

The structural system equations of motion are thus transformed to the general Duffing-

type reduced modal equations with constant nonlinear modal stiffness matrices.

The effect of flow yawing on the critical flutter dynamic pressure for isotropic

and orthotropic rectangular panels at supersonic speeds was investigated in the late sixties

and early seventies. Kordes and Noll [41], and Bohon [42] have theoretically studied the

influence o f arbitrary flow angles on isotropic and orthotropic rectangular panels with

classical simply supported boundary conditions. Durvasula [43, 44] used the Rayleigh-

Ritz method and 16-term beam functions to study the flow yawing and plate obliquity

effects o f simply supported and clamped rectangular isotropic panels. Kariappa et al. [45]

and Sander et al. [46] used the finite element method to study the effects of flow yawing

of isotropic parallelogram panels. The dependence of critical dynamic pressure on the

flow angle and flexible supports has been shown experimentally and theoretically by

Shyprykevich and Sawyer [47] and by Sawyer [48]. It was found that orthotropic panels

mounted on flexible supports experience large reductions in critical flutter dynamic

pressure for only small changes in flow angle.

An exhaustive search of the literature reveals that there are very few

investigations on nonlinear panel flutter considering the effects o f flow yawing.

Friedmann and Hanin [49] were the first to study supersonic nonlinear flutter of

rectangular isotropic and orthotropic panels with arbitrary flow direction. They used the


7

first order piston theory for aerodynamic pressure and Galerkin’s method in the spatial

domain to analyze nonlinear panel flutter with yawed supersonic flow. The reduced

coupled nonlinear ordinary differential modal equations were solved with numerical

integration. Using a 4x2 modes model, four natural modes in the x-direction and two

modes in the y-direction, LCO were obtained for simply supported isotropic and

orthotropic rectangular panels. Chandiramani et al. [50] used the third-order piston theory

and Galerkin’s method in the spatial domain. The reduced coupled nonlinear ordinary

differential modal equations were solved using a predictor and a Newton-Raphson type

corrector technique for limit-cycle periodic solutions. Direct numerical integration was

employed for nonperiodic and chaotic solutions. A 2x2 modes model, two natural modes

in the x- and y-directions, was used for simply supported rectangular laminated panels.

Abdel-Motagaly et al. [51] have recently extended the finite element method to study

nonlinear flutter of composite panels with yawed supersonic flows using the MIN3

triangular element developed by Tessler and Hughes [52] and extended for nonlinear

analysis by Chen [53]. It was found that, for laminated composite panels [54], the flow

direction could greatly affect the limit-cycle behavior.

In addition to the aerodynamic loading, aircraft and spacecraft panels are

subjected to high levels o f acoustic loading (sonic fatigue), due to high frequency random

pressure fluctuation. A comprehensive review of sonic fatigue technology up to 1989 is

given by Clarkson [55] where various types of pressure loading, developments of

theoretical methods, and comparisons of experimental and analytical results were given.

Recently, Wolfe et al. [56] gave a review of sonic fatigue design guides, classical and

finite element approaches, and identification technology including experimental

investigation of nonlinear beams and plates response. Sonic fatigue design guides based

on test data and simplified single mode solutions were given by Rudder and Plumblee

[57] for isotropic panels and by Holehouse [58] for composite panels. A Solution method

based on Galerkin procedure and on time domain Monte Carlo approach was developed

by Vaicaitis [59] for nonlinear response of isotropic panels under acoustic and thermal

loads. Composite panels were considered by Arnold and Vaicaitis [60], and by Vaicaitis

and Kavallieratos [61], Bolotin [62] used the Fokker-Planck-Kolmogorov (FPK) exact

method to solve single DOF forced Duffing equation. The finite element/equivalent


8

linearization method was used by Chiang [63] to analyze the large deflection random

response o f complex panels.

Sonic fatigue and panel flutter have been independently considered for aircraft,

spacecraft, and missiles. However, up to very recently there was no study for nonlinear

panel response under combined acoustic and aerodynamic loading. Abdel-Motagaly et al.

[64] presented a study for nonlinear composite panels response under combined acoustic

and aerodynamic loading, which is based on the research presented in this thesis.

1.2.2 Piezoelectric Sensors and Actuators

Since the discovery of piezoelectricity by the Curie brothers in 1880 [65], there

have been many applications in various fields using piezoelectric materials, such as

ultrasonic transducers, telephone transducers, and accelerometers. Piezoelectric materials

basically convert mechanical energy to electrical energy and vice-versa. When

mechanical force or strain is applied to a piezoelectric material, electrical charge or

voltage is generated within the material, this is known as the direct piezoelectricity effect.

Conversely, when electrical charge or voltage is applied to the piezoelectric material, the

material generates mechanical force or strain, this is known as the converse

piezoelectricity effect. The piezoelectric direct and converse effects are the basis for

using them as sensors and actuators, respectively. The linear piezoelectricity constitutive

relations that relate the mechanical and electrical variables for linear material behavior

are given by [65]:

where {cr], [ f] , [D ], and {£} are stress, strain, electrical displacement, and electrical

field, respectively, [<2]£ is the piezoelectric stiffness matrix at constant electrical

field, is the dielectric permittivity matrix measured at constant strain, and [e] is the

piezoelectric electro-mechanical coupling constants matrix. Based on the principle o f

virtual work, a useful analogy between piezoelectric electrical and mechanical variables

can be determined as given in Table 1.2. For example, the electrical field applied or

sensed in the piezoelectric material is analogous to mechanical strain.


9

Although piezoelectricity was discovered a long time ago, the application of

distributed sensing and actuation using piezoelectric materials for flexible structures is

relatively new. Bonding or embedding piezoelectric sensors and actuators to flexible

structures allows for measuring and applying mechanical strains and consequently

suppressing undesired structure vibrations, hence improving the life duration and

performance of the structure. Figure 1.2 shows a typical piezoelectric element that could

be used as sensors or actuators. Traditional isotropic piezoelectric material is usually

manufactured from lead zirconate titanate (PZT) or polyvinylidene fluoride (PVDF)

piezoelectric ceramics. Piezoelectric properties are induced in the ceramics using the

polling process during which a high dc electrical field is applied to the ceramic in a

specific direction.

Recently, many articles dealing with piezoelectric sensors and actuator modeling

for active structure vibration have appeared in the literature. A review article of the

applications and modeling of distributed piezoelectric sensors and actuators in flexible

structures up to 1994 is given by Rao and Sunar [65]. A general review for intelligent

structures including piezoelectric sensors and actuators is given by Crawley [66]. The

governing equations for piezoelectric sensors and actuators using the classical approach

were considered by many authors [67-69], This research is concerned with the modeling

of piezoelectric sensors and actuators embedded in composite panels using the finite

element method. The first article for modeling piezoelectric continua using finite element

was given by Allik and Hughes [70], where they formulated finite element equations for

piezoelectric continua based on linear piezoelectric constitutive relations and using an

isoparametric tetrahedral element. Tzou and Tseng [71] developed a new thin

piezoelectric solid finite element with internal degrees of freedom that is more suitable

for modeling distributed piezoelectric sensors and actuators in plate and shell structures.

Ha et al. [72] used an eight-node three-dimensional composite brick element to model

dynamic and static response of laminated composites containing piezoelectric sensors and

actuators. Hwang and Park [73] used Hamilton’s principle to derive the equations of

motion of a laminated plate with piezoelectric sensors and actuators. They used a new,

two-dimensional, four-node, 12 degrees of freedom quadrilateral plate bending element

with one additional electrical degree of freedom to eliminate the problems associated


10

with using solid elements and to reduce the size of the finite element equations. A

conforming rectangular plate element based on classical plate theory was also developed

by Zhou [74] to model composite panels with piezoelectric actuators. This formulation

was used for panel flutter suppression analysis. The same element was used by Liu et al.

[75] for vibration control of composite plates. Suleman and Venkayya [76] used a 4-node

bilinear Mindlin plate element with additional electrical degree of freedom. Detwiler et

al. [77] modified the QUAD4 isoparametric quadrilateral element to handle laminated

composite plates containing piezoelectric sensors and actuators. Sze and Yao [78] used

and compared the performance of various solid shell and membrane elements to model

surface bonded piezoelectric patches. Recently, Bevan [79] modified the shear

deformable MIN6 shell element to model composite shell structures integrated with

piezoelectric sensors and actuators.

One new concept for piezoelectric sensors and actuators that is utilized in this

research is the self-sensing piezoelectric actuators introduced by Dosch et al. [80] and by

Anderson and Hagood [81]. This concept combines the sensing and actuation functions

into a single piezoelectric piece through the use o f an electrical circuit that measures the

sensing charge output of piezoelectric actuators. The use of such concept allows for

collocated sensing and actuation, which is a preferable property for active vibration

control. Another new concept is the use of anisotropic piezoelectric actuators with

interdigital electrodes, such as the active fiber composites (AFC) piezoelectric actuator

[82], and the Macro-Fiber Composite (MFC) piezoelectric actuator [83]. For both AFC

and MFC, the polling and excitation fields run parallel to the plane of actuation compared

to vertical to the plane of actuation for traditional piezoelectric materials. This permits the

use of the more efficient “33” piezoelectric coupling constant which is usually twice the

value of the traditional piezoelectric “31” and “32” coupling constants (see [84] for

detailed electrical and mechanical properties of piezoelectric ceramics).

One important problem when using distributed piezoelectric sensors and actuators

for active structural control is the optimal placement of sensors and actuators. Some

examples for the methods used in the literature for optimal actuator and sensor placement

are given in [85-92], Some of the methods used are based on open loop criteria such as

maximum controllability and observability [85, 86], Another class o f methods is based on


II

minimization of a linear quadratic regulator (LQR) cost function using gradient

optimization methods [87-90]. A third class of methods is based on using more rigorous

optimization methods such as the genetic algorithm [91] and gradient based [92]

optimization techniques. More details for piezoelectric sensor and actuator placement for

the problem of panel flutter suppression will be discussed in the next subsection.

1.2.3 Panel Flutter Suppression Using Piezoelectric Materials

Many researchers have investigated the effectiveness of using piezoelectric

materials for passive or active control of flexible structures. However, only few studies

have been reported for linear and nonlinear supersonic panel flutter suppression using

piezoelectric materials. Scott and Weisshaar [93] were the first to study the suppression

of linear panel flutter using piezoelectric materials. The piezoelectric materials covered

the full surface of the panel and were used to generate bending moments to control panel

flutter. Four modes were retained using the Ritz method, and the panel was modeled as a

simply supported isotropic plate. Linear optimal control theory using full state feedback

LQR was employed in the simulation. Hajela and Glowasky [94] applied piezoelectric

elements in linear panel flutter suppression. Finite element models for panels with surface

bonded and embedded piezoelectric materials were generated to determine the response.

The actuation forces generated by the piezoelectric material were incorporated as static

prestress in the finite element models. Using a multi-criterion optimization scheme, the

optimal panel configuration with minimum weight and optimal sizing and layout of the

piezoelectric elements for maximum flutter dynamic pressure were determined. Using a

finite element approach, Suleman and Goncalves [95], and Suleman [96] recently

investigated a passive control methodology for linear panel flutter suppression. The

methodology induces tensile inplane loads from bonded or embedded piezoelectric

patches to increase panel critical dynamic pressure. They proposed the use of the physical

programming optimization method to determine optimal actuator configuration. Surace et

al. [97] used piezoelectric sensors and actuators to suppress linear supersonic panel flutter

using robust control techniques based on structured singular values for a simply

supported composite panel over a range of Mach numbers. The panel was modeled using

Galerkin’s method with classical plate theory and linear piston theory for aerodynamic

loading. Frampton et al. [98] employed a collocated direct rate feedback control scheme


12

for the active control of linear panel flutter. The linearized potential flow aerodynamics

was used for the full transonic and supersonic Mach number range. They demonstrated

that a significant increase in the flutter boundary was achieved for a simply supported

square steel panel.

The first study of nonlinear panel flutter suppression using PZT piezoelectric

actuators was given by Abou-Amer [99]. He used piezoelectric layers to generate inplane

tension forces and consequently increase the panel flutter boundary. He showed that the

PZT material is more capable of preventing nonlinear panel flutter compared to using

active constrained layer damping. Lai et al. [100-102] studied the control of nonlinear

flutter of a simply supported isotropic plate using piezoelectric actuators. The Galerkin’s

method was adopted in obtaining the nonlinear modal equations. The optimal control

theory and numerical integration were used in the simulation. They concluded that the

bending moment induced by piezoelectric actuators is much more effective than inplane

forces for flutter suppression. Zhou et al. [103, 104] and [74] used the finite element

method to control isotropic and composite panels with surface bonded or embedded

piezoelectric patches. The finite element formulation considered coupling between

structural and electrical fields. An optimal full state feedback LQR controller was

developed based on the linearized modal equations. The norms of the feedback control

gain (NFCG) were used to provide the optimal shape and location of the piezoelectric

actuators. Numerical simulations showed that the critical flutter dynamic pressure is

increased about four times and two times for simply supported and clamped isotropic

panels, respectively. Dongi et al. [105] have presented a finite element method for

investigations on adaptive panels with self-sensing piezoelectric actuators. The

LUM/NTF algorithm was extended to include the linear and nonlinear active stiffness

matrices due to output feedback. A control approach based on output feedback for active

compensation o f aerodynamic stiffness (ACAS) terms is developed. They showed that

the ACAS control is able to increase the linear flutter boundary Mach number from =

3.22 to M e = 6.67 for a simply supported isotropic panel. Wind tunnel testing performed

by Ho et al. [106] has shown that panel limit-cycle motions observed in the wind tunnel

can be successfully reduced for composite panels with one-sided surface mounted

piezoelectric actuators and strain sensors using an iterative root locus based gain tuning


13

algorithm. Their wind tunnel testing showed the leading edge piezoelectric actuator

patches to be more effective than the trailing edge patches in suppressing panel flutter.

Very recently, Kim and Moon [107] presented a comparison between active control and

passive damping using piezoelectric actuators for nonlinear panel flutter. The finite

element method was used to model the panel and LQR control method was used for

active control. The shape and location of the piezoelectric actuators was determined using

genetic algorithms.

With this exhaustive search for panel flutter suppression studies, two main

findings are determined. First, the effect of flow yaw angle has never been considered in

the literature for both linear and nonlinear panel flutter suppression, despite its great

effect on the panel flutter mode shape and, consequently, on the optimal location of

piezoelectric actuators and sensors. Second, most of the studies used LQR full state

feedback control assuming that all the states are available without any consideration for

the problem of state estimation for the nonlinear system dynamics or used non-optimal

output feedback based on iterative design methods.

1.3 Outline of the Study

This study presents multi-disciplinary research that includes nonlinear finite

element modeling of composite panels, aeroelasticity, modeling and optimal placement of

piezoelectric sensors and actuators, and control theory. It could be divided into two parts.

The first part covers finite element modeling and analysis for nonlinear flutter of

composite panels considering the effect of flow yaw angle and the effect o f additional

high acoustic pressure loading. The second part covers nonlinear flutter suppression for

composite panels under yawed supersonic flow using piezoelectric sensors and actuators

including optimal sensor and actuator placement and controller design.

The thesis is organized as follows. In Chapter 1, background material and

literature survey are given for the main topics of this research including nonlinear panel

flutter analysis, piezoelectric sensors and actuators, and panel flutter suppression

methodologies followed by an outline of the thesis contents. The derivation of finite

element coupled nonlinear equations of motion for composite panels with integrated

piezoelectric sensors and actuators under yawed supersonic aerodynamic flow and

acoustic loading is presented in Chapter 2 using the three-node triangular MIN3 plate


14

element with improved transverse shear [48]. The MIN3 element is modified to handle

piezoelectric sensors and actuators by using an additional DOF for electrical potential per

each piezoelectric layer. In Chapter 3, modal transformation and reduction, and solution

procedure for nonlinear panel response are presented. The system governing equations

are transformed into the modal coordinates using the panel linear vibration modes to

obtain a set o f nonlinear dynamic modal equations of lesser order that can be easily used

to solve the problems of linear and nonlinear flutter boundaries and to analyze panel

response under combined acoustic and aerodynamic pressures. The reduced modal

equations o f motion are also used to design control laws and to simulate panel flutter

suppression. Solution procedures based on time domain numerical integration methods

and based on frequency domain methods for nonlinear panel flutter are also described in

this chapter. Validation o f the MIN3 finite element modal formulation and analysis

results are presented in Chapter 4. The MIN3 finite element modal formulation is

validated by comparison with other finite element and analytical solutions. Analysis

results for the effect of arbitrary flow yaw angle on nonlinear supersonic panel flutter for

isotropic and composite panels are presented using the frequency domain solution

method. In addition, the effect of combined supersonic aerodynamic and acoustic

pressure loading on the nonlinear dynamic response of isotropic and composite panels is

presented. Description of the different control methodologies used for nonlinear panel

flutter is presented in Chapter 5. Optimal control strategies [104] are the main focus for

the suppression of nonlinear panel flutter in this study. The linear quadratic Gaussian

(LQG) control, which combines both linear quadratic optimal feedback (LQR) and

Kalman filter state estimator, is considered as systematic linear dynamic compensator. In

addition, extended Kalman filter (EKF) for nonlinear systems [105-108] is also

considered and combined with optimal feedback to form a nonlinear dynamic output

compensator [109]. Finally, a more practical approach based on optimal output feedback

is used. Closed loop criteria based on the norm of feedback control gains (NFCG) for

actuators and on the norm of Kalman filter estimator gains (NKFEG) for sensors are

described to determine the optimal location of self-sensing piezoelectric actuators.

Simulation studies for the suppression of nonlinear panel flutter using piezoelectric

material under yawed supersonic flow are presented in Chapter 6 . Comparison of the


15

different controllers considered is performed to determine the effect o f different control

strategies on the panel flutter suppression performance. In addition, results for nonlinear

panel flutter suppression under yawed supersonic for a specific range o f yaw angles,

including optimal actuator and sensor location, are presented for both isotropic and

composite panels. In Chapter 7, summary, conclusions, main contributions, and

recommendation for future work are given.


16

Table 1.1 Panel Flutter Theories

Type Structure Theory Aerodynamic Theory Range of Mach No.

1 Linear Linear Piston V2 < Moo < 5

2 Linear Linearized Potential

Flow

1 < Moo < 5

3 Nonlinear Linear Piston y f l < Moo < 5

4 Nonlinear Linearized Potential

Flow

1 < Moo < 5

5 Nonlinear Nonlinear Piston Moo > 5

Table 1.2 Analogy between electrical and mechanical

variables for piezoelectric materials

Mechanical Electrical

Displacement u (vector) Electric Potential 0 (scalar)

Stress cr(2nd order tensor) Charge flux density (Electrical

Displacement) D (vector)

Strain f ( 2 nd order tensor) Electric field E (vector)


Wni

ax/h

Flutter analysis Noise analysis

0.6

0.4

Dynanic Pressure, q

(4 , 1)

(0

mode (2,1)

mode (1,1)

Dynanic Pressure

Panel response Mode coalescence

A irflow

Flutter deflection mode shape for simply supported panel

i

0.5

0

0.5

1450 460 470 4 80 490 500

Tim e (msec)

Flutter limit-cycle oscillation

Figure 1.1 Explanation of nonlinear panel flutter phenomenon


Polling direction

Applied or Sensed Electrical Field >

Surface electrodes

Figure 1.2 Schematic of traditional isotropic piezoelectric element


19

CHAPTER II

FINITE ELEMENT FORMULATION

2.1 Introduction

The three-node triangular Mindlin (MIN3) plate element with improved

transverse shear, developed by Tessler and Hughes [52], is used in this study. This

element has Five degrees o f freedom per node and it uses a special interpolation scheme,

anisoparametric interpolation, to avoid the problem of shear locking commonly arising

when using the standard isoparametric interpolation approach. Additionally, an element-

appropriate shear correction factor is used to enhance element transverse shear energy.

Due to these improvements, the MIN3 element produces a well-conditioned element

stiffness matrix over the entire range o f thickness to length ratios. Based on extensive

numerical testing of the MIN3 element, Tessler and Hughes concluded that this element

is an excellent element for linear problems and is a very viable candidate for laminated

composites and nonlinear problems. Furthermore, Chen [53] demonstrated the efficiency

of this element for nonlinear problems by using it for the analysis of nonlinear panel

response under thermal and acoustic loading.

The MIN3 element is modified to handle piezoelectric sensors and actuators by

using an additional DOF for electrical potential per each piezoelectric layer. By doing

this, the modified MIN3 element becomes a fully coupled electrical-structure composite

plate element. The following are the main assumptions used in the formulation:

• Thin skin panels with embedded or bonded piezoelectric layers.

• Bending theory of Mindlin (first order shear deformation theory).

• Composite laminate theory.

• First order piston theory is used to model aerodynamic pressure for yawed

supersonic flow (1 .6 < Moo < 5).

• Large deflection effect is considered using nonlinear von-Karman strain

displacement relations.

• Linear constitutive relations for mechanical-electrical coupling are used

(linear piezoelectricity).


2 0

In the following sections, the detailed derivation of the nonlinear dynamic equations of

motion for a laminated composite plate with embedded or bonded piezoelectric sensors

and actuators is given. The loads considered are aerodynamic pressure due to supersonic

yawed flow, random acoustic loading, and piezoelectric electrical loading (see Figure

2.1). This general system of coupled nonlinear equations will later be used to analyze

nonlinear panel flutter under yawed supersonic flow and nonlinear panel response under

combined aerodynamic and acoustic loading. In addition, these governing equations form

the basis for the design and simulation of control system design for nonlinear panel flutter

suppression.

Based on the Mindlin plate bending theory, the element displacement functions

are given by:

u x = u ( x , y , t ) + zip v (.v, y j )

u . = w ( x , y , t )

where u x, u y , u z are the displacement components at any point within the element; u, v, w

are the displacements o f the plate mid-plane; and y/x, i//y are the rotations of the mid-plane

normals due to bending only.

The electrical potential DOF is assumed constant for each piezoelectric layer; i.e.

w 0 is assumed constant over the element area.

2.2 Element Displacement Functions

u v = v(.v, y , / ) + z y / x (x, y , t ) ( 2 . 1)

w 0 (.v ,y , z , t ) = w tp ( z k , [ )

where Zk is the z coordinate o f the k lh piezoelectric layer.

Then the element degrees of freedom are defined as:

(2 .2 )

(2.3)

(2.4)


2 1

where V is the electrical potential (voltage) for each piezoelectric layer and n p is the total

number o f piezoelectric layers per element (see Figure 2.2).

Displacement fields over the element are expressed in terms of element nodal

DOF and element interpolation functions as follows:

w(.v, y , 0 = 1H w J{wfc } + \_H w ¥ J{ p }

if/x ( x , y , t ) = [ / / « * - J M

iffy (x , y j ) = \ H m , ] f y / } (2 .5 )

u { x , y j ) = \ H u J {w „ ,}

v(.v, y , t ) = \_HV }

The element interpolation functions are expressed in terms of element area coordinates

(see Figure 2.3), and the element quadratic interpolation polynomials as given by

equation (2 .6 ).

f t f t j

f t l , m , m 2 Jl / v J = L « , , J = l i i * f t o o oj l/vJ=Ltf..J=L<> o o * f t f t j

Element area coordinates, 4i. are related to the element geometric coordinates using the

following transformation:

1 ~ l i i ' ’4 i.V • = x \ x 2 -r3 %2

. y . , y \ y 2 >’3.' ^ i i

1~ 2 A

x2>’3 - -v3-v2 .v2 - 3*3

rri1CO F

x 2 y i - ^ 1^3 V3 “ Vi x l ~ x 2 X

.^3 . .-vi -v2 ~ x 2 y \ -V1 _ .v2 x 2 ~ -VI . Vv. - J

(2.7)

where (x„y,) are the coordinates o f node /, and A is the triangular element area given by:

A = ^ ( x 2 —-vi )(y3 ~ >"i) — (-r3 ~ x 0 ( y i “ Vi)-

The element interpolation polynomials are defined as:


2 2

L x ~ — ( b 2 N 4 - b 2 N 6 ), L 2 = ~ ( b \ N 5 - b 2 N 4 )o o

L 2 = —( b 2 N 6 - b i N 5 ), M { = — ( a 2 N 6 - a 2 N 4 ) o o

M 2 = - ( a 2 N 4 - a { N $ ), M 3 = — ( a [ N 5 - a ^ i N f r ) 8 8

(2 .8 )

N 4 - 4 ^ \ q 2 < (V5 “ 4^2C3- ^ 6 = 4c3C[a { = x 2 - x 2 , a 2 = x x- x 3, = .v2 --V!

b \ = ,v2 ~ y 3 ’ b 2 = >*3-Vi, b 2 = >’i -V2

2.3 N onlinear S train-D isplacem ent Relations

The von-Karman large deflection strain-displacement relations for a plate

undergoing extension and bending at any point - through the plate thickness are given by:

{*}= {e" }+ ;{*}= y „ }+ { e " }+ ;{*}= { 4 }+ i [ e ] { S } + c M (2.9)

£ } = ■£ xe y - & , } =

u , xV.y ■. M = -

V y . x

V x . y

V X V u , Y+ v , x V x . x + V y . y

[«] =W , x 00 vv, v

IV, v VV’, r

(2 . 10)

vv,,VV.,

where denotes derivative with respect to the subscript. The shear strain-displacement

relations are:

y yz w , v V x- = ■

- M >

Yxz v v \ r V vI '

( 2 . 11)

Substituting the MIN3 element interpolation functions, equation (2.5), the strain

components in equations (2.9) and (2.11) can be expressed in terms of element nodal

coordinates as follows:

& ^Pif/b ]{lvfc \p y / y / ]{(^}

M = [ Q J M

{ / } - \C y b ]{wfc }+ \ C y V ]{^}

(2 . 12)


where the strain interpolation matrices are given by equations (2.13) through (2.16).

[C, „ ]= L H V ± y

.L^vJvv+L^./ J’V(2.13)

(2.14)

(2.16)

(2.15)

Expressions for these strain interpolation matrices as functions o f element geometry and

area coordinates are given in Appendix A.

Based on the analogy given in chapter 1 between electrical and mechanical

quantities for piezoelectric materials, the electrical field-potential relations are analogous

to mechanical strain-displacement relations. Assuming that the electrical degrees of

freedom are constant over each piezoelectric layer in the element, the electrical field is

related to the electrical DOF by:

Both traditional isotropic piezoelectric materials and anisotropic piezoelectric actuators,

such as active fiber composite (AFC) [82] and macro-fiber composite (MFC) [83], are

covered by the presented formulation. For isotropic piezoelectric materials, the

polarization direction is the ‘"3” direction; consequently, {E,} is the electrical field in the

3 direction and /i( is the thickness of the piezoelectric layer. For anisotropic piezoelectric

2.4 Electrical Field-Potential Relations

(2.17)


24

actuators such as MFC or AFC, the polling direction is the “ 1” direction. For this case,

{£,} is the electrical field in the 1 direction and h i is the electrode spacing.

2.5 Linear Piezoelectricity Coupling Equations

The stress, strain, electrical field, and electrical displacement within a

piezoelectric layer can be fully described by the following linear electromechanical

relationships [65]:

(, Ig){£>}=[e]£r}+[e]f {£}

where \q Y " is the piezoelectric layer stiffness matrix at constant electrical field, is

the permittivity matrix measured at constant strain (clamped), [e] is the piezoelectric

constant matrix. In practice, the constants [e] and [ e ] f may be unavailable. They are

expressed in equation (2.19) in terms of permittivity at constant stress [e]*7 , and the

piezoelectric constant matrix [ d \ , which are more commonly available

[ e ] = [ d ] \ Q ) E(2.19)

[e ] f = [ G ] a - [ d ] [ Q ] E [ d ] T

By substituting (2.19) into (2.18), the linear electromechanical coupling constitutive

relations for mechanical stress and electrical displacement can be written as:

M = [ f i ] £ ( fr H r f]r {E}) (, 90){ D } = fc /]M + fe ]CT{£}

The electrical displacement, {D}, in these equations represents electrical charge flux

density and is analogous to mechanical stress. For an isotropic piezoelectric element with

its poling axis in the “3” direction, the matrix of piezoelectric constants [r/] has the form:

' 0 0 0 0 d \ 5 00 0 0 d \ 2 0 0

.^31 ^32 d 33 0 0 0

The first term in the subscript refers to the axis of applied electrical field and the second

subscript refers to the axis of resulting mechanical strain. For thin piezoelectric ceramics,

< 33 = d is =0 and d j i = <32. For the case of anisotropic piezoelectric, MFC or AFC, with


25

the poling axis in the “ 1” direction, the constants d - u and d 32 are replaced by d \ \ and d \ 2

which are determined experimentally [83].

2.6 Constitutive Equations

Lamina stress-strain relations are derived based on the Mindlin plate theory and

on the linear electromechanical piezoelectric coupling equations. It is assumed that all

laminas are perfectly bonded with zero glue thickness. For a laminated fiber reinforced

composite panel with embedded or bonded piezoelectric layers, the kth layer stress-strain

relationships in the material principal axes (1,2,3) are given by:

(2 .22 )0-1 Qw 012 0 n

/’ £\ d i\

\O’ 2 • = 012 0T> 0 * e2 ' - E a t - d i2 •

r12. k 0 o' 066. k y 12 0

The klh layer could be either structure lamina (£,* = = 0) or piezoelectric layer. The

transverse shear stress-strain relations are also given by:

^ 23! _r13 J*

<244 0 . 0 055

V23

k(2.23)

For regular isotropic piezoelectric layers i = 3, corresponding to applied or sensed

electrical field in the “3” directions while for MFC or AFC piezoelectric actuators / = 1,

corresponding to applied or sensed electrical field in the *T” direction.

In laminate reference axes (x, y, z), the constitutive equations are:

(2.24)' O n 012 016

/d x

\

• = 012 022 026 £ y ~ E ' k ’ d y •

k 016 026 066 k Y x \ d xy k >

| r .v: 044 045' K \l r -vc.K = 045 055. t 1

(2.25)

(2.26)

In matrix compact format these equations can be written as:

{<7}t = E l ({<?}-£,*{<*},)

where, \ Q ] and \ Q S ] are the lamina transformed reduced stiffness matrices for plane

stress and transverse shear respectively, and { d } is the transformed piezoelectric


26

constants vector. Details of the derivation of transformed stiffness and piezoelectric

constants using laminated composite theory are given in Appendix B.

In addition to mechanical stress, the electrical displacement o f piezoelectric

layers, given in equation (2 .20), can be written as:

D« = ¥ 1 ( t?} - E i t ¥ \ E ik (2.27)

Thus, equations (2.26) and (2.27) represent the constitutive relations for a general

composite lamina or a piezoelectric layer.

2.7 Laminate Resultant Forces and Moments

In a general composite laminate, stress is different from one layer to another. The

resultant stresses for the laminate, or forces and moments per unit length, are obtained by

integrating stress over the laminate thickness, h, as:

h! 2({w }{M })= J{cr}t ( I , : ) *

- h i 2hi 2

{*}= J H d z

(2.28)

- h t 2

Using the lamina constitutive relations, the force and moment resultants are written as:

"A B ~ £ ° N 0

B D . K A V (2.29)

where the laminate extension, extension-bending, bending, and shear stiffness matrices,

[A], [£], [ D ] , [A5], are defined as:

/i/2

(M,[fl].[0]) = J leU ,:,:2}/;—h i 2

h / 2 _[ A J = \ \ Q s \ d z

- h / 2

and the piezoelectric resultant force and moment are:

h / 2 _

( K ) . W } ) = \ \ Q \ { d } k E l t ( l , z ) d z- h / 2

(2.30)

(2.31)


27

These force and moment resultants can be expressed in terms of element nodal DOF

using equation (2 . 12) for { ? } and { k }, and equation (2 .17) for E ik as follows:

{W}= [A][C„, ]{«-„ }+2- M[0]( [c*, ] k }+ [c„ ]fcr})+ [B}[Cb }

{M}= [J][C.]{,, }+Y&ttajflc*, I k }+ [c„ ]M)+[o][c,, }{«}= [ a , j[c ]k }+ [4S ][cw ]K

(2.32)

and the resultant piezoelectric force and moment as:

K F - k -i [bJ K IV }= ~ \-P m ]

where the piezoelectric matrices [TV] and [ P m ] are defined as:

KJ=[lcl{rf},A, - IQ\{d\hk ... \5 lp{d}„ph,v ]

= ... E l - K K w - ; * ) 2 (2.34)

[i2 \ i p I — )” J

2.8 Aerodynamic Pressure Loading

Assuming an airflow that is parallel to the panel surface, the aerodynamic

pressure loading is expressed using the first-order piston theory. This theory relates the

aerodynamic pressure and panel transverse deflection as follows:

(2.33)}= 1 feJW }

A P a = -2 qa

J3' . M i - 2 1 A

vv, r c o s a + w , .. s in a -i ---------------w .rM i - l V „

(2.35)

where q a = p a V z /2 is the dynamic pressure, p a is the air density, V „ is the airflow

velocity, vv is the panel transverse displacement, is the Mach number, a is the flow

yaw angle, and /? = y j M i - 1 . Using non-dimensional parameters, equation (2.35) can

be written as:

A — [vv, v cos a + w, v sin a ] + — l~ w, tA P a = “

where

f r> - _ „ r » \(2.36)

a 3 o a 4 j


28

, _ 2 q a a Z _ p a V x { M l - 2 )Rr\ ’ 3 a ~P D 110 p h ( D 0 ( j

(2.37)

are the non-dimensional dynamic pressure, non-dimensional aerodynamic damping,

aerodynamic damping coefficient, and panel reference frequency, respectively. Duo is the

first entry of the laminate bending stiffness matrix [D] determined with all fibers of the

composite layers are in the x-direction as a reference, p , a , and h are the panel density,

length and thickness, respectively.

Equation (2.36) shows that the aerodynamic loading on the panel is function of

both local panel slopes in the .v and y directions and of the panel vibration velocity;

therefore panel flutter is a self-excited vibration. Using the MIN3 element interpolation

functions, the aerodynamic pressure can be expressed in terms of element nodal DOF as

follows:

2.9.1 Generalized Hamilton’s Principle

Element matrices and nonlinear coupled electrical and structural equations of

motion for a composite laminated plate element with piezoelectric layers are derived

using the generalized Hamilton’s principle. In addition to aerodynamic loading, a random

surface pressure is also considered to allow the study of panel response under combined

aerodynamic and acoustic loading. The generalized Hamilton’s principle is:

where T is the kinetic energy, U is the strain energy, W eiec is the electrical energy, and

W e.xt is the work done by externally applied forces and electrical voltage. These energy

terms are defined by the following volume integrals:

(2.38)

( L t f » J K } + ! « » • * > } )2.9 Element Equations of Motion and Matrices

(2.39)

(2.40)Vol “


29

^ = J \ { e } T (2.41)Vol ~

K l « = / (2.42)Vol

The work done by external forces and voltage is given by the surface integrals:

W e x t = J iw7 iFs } I S - J (2.43)5, S2

where {Fs} represents the surface loading on the element due to aerodynamic pressure

and acoustic pressure, 5/ is the area of the surface loading. V is the applied voltage, p iS is

the surface charge density of the piezoelectric layer, and S 2 is the area of piezoelectric

layer. All variations at times // and t 2 must vanish in the Hamilton’s principle, and thus it

can be written as:

V° l (2.44)+ { S e Y {£> } ]r/V + J {<5W}r { F s y i s - J S V p c s d S = 0

Si s 2

The element matrices are determined by evaluating equation (2.44) term by term to arrive

at the element nonlinear dynamic equations of motion.

2.9.2 Element Stiffness and Electromechanical Coupling Matrices

Element stiffness matrices including piezoelectric electromechanical coupling

terms are derived using the variation of strain energy and electrical energy terms.

S t r a i n E n e r g y :

Using the definition of stress force and moment resultant, the strain energy variation term

can be written as:

S U = 7 {^1+ { S f c f { M }+ a s {S y f (2.45)

where a s is the MIN3 element shear correction factor, which is defined as function of the

ratio of the diagonal shear and bending stiffness coefficients associated with the

rotational degrees of freedom:


The exact definition of element shear and bending stiffness matrices will be given later.

More details about the selection of the shear correction factor are given in [52].

Substituting for strain and stress resultants in terms of element interpolation matrices and

element nodal DOF into equation (2.45), the strain energy variation can be expressed as:

*/ = J {({&■„, }r [c,„ f + {&■„ }r [cVi, r rnT + {Svf [c„ F l e f )

[A][c,„ ]{»•„, } + - [A ][s ] [c f* ] K }

+ {[A][0][cw ]fr}+[S][C (, ]{(ir}+ [/>* ][«„ I K }]

(2.46)

+

^[B][<9][C^ ]{¥ }+ [D][Cb ]{¥ }+ [PM ] [B0 ]{w0 }

as ({ v’6 }T \Cyb Y + {V}r \cw F )(U, ] [Cyt ]{wb }+ U , ] [Cw ]{¥ }) }dA

(2.47)

(2.48)

Electrical Energy:

By substituting equation (2.27) for electrical displacement, the variation of electrical

energy can be written as:

r i i /2 t ~ \ "■ 5 W W = J £ t t )</: (2.49)

A L-/j / 2

Using the definition of piezoelectric matrices [/V] and [Pa/], the electrical energy

variation becomes:

S W elei. = J { 5 E ( }r ( [ P , v f f M + K I K { £ / } ) < « ( 2 .5 0 )

where the new piezoelectric matrix [P<,] is defined as:


31

0 e ?^ Itfip

0

- U V „ P \ Q l p V htn p

(2.51)

Using the strain-displacement relations, the electrical energy variation can be expressed

in terms of nodal DOF as follows:

= j l s T K F [Jvf [cm]{wm}+-[e][cvb]{w6}A 1 ^

+ ^ [e ] [c „ » ,]M )+ [P M f [ Q , ] M + M K } l d A

(2.52)

J J

The linear and nonlinear element stiffness and electromechanical coupling matrices can

be found now from the variation of both strain and electrical energies by writing the sum

of S U and S W eiec in matrix form as follows:

S w b T '0 0 08 y / 0 [ * v }\pni

S w ni*

0 [ k \n i tp [ k ] m

S w 0K

0 \.k]<pip \-k \0m

0 [ ^ ' i h ip [ ^ 1 h m [ * l k

[ ^ 1 \ipb [*iV [^•1 \tpm U'lV[^■11 mlp 0 0

}<pb [ ^ • i h ip 0 0

o n [ k s ]b V 0 0 "

[k] <P0+ a . [ ^ • s 1 ipb [ k s 1 \p 0 0

+[k] m 0

j0 0 0 0

[ k ] 0 _ 0 0 0 0

" [* . v 0 h \-k\N 0 hip 0 o '

+ \.k\tf(t) \yjb r * i w V 0 0+

0 0 0 00 0 0 0

(^•1 Nm h h \p 0 o ' \ - k \N b h tp 0 o '

t^-' l Nm Jipb h 0 0+ tpb \-k\N iA ip 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 _

+

~ i k 2 \b \.k2 h ip 0 0 " >w b

[ k 2 ]ipb [ k 2 \v 0 0 ¥

0 0 0 0 w m0 0 0 0

/w<p

= S U + d W elec (2.53)

where the element sub-matrices in equation (2.53) are defined by the following equations.


32

L i n e a r s t i f f n e s s s u b - m a t r i c e s :

[ k ] w = \ [ C b ]T [ D ] [ C b ] d A (2.54)<4

= \ [ C b ]T [ B ] [ C m ) d A = [ k \ Tm ¥ (2.55).4

[*]„, = j [ C m f [ A ] [ C m ]clA (2.56)A

P i e z o e l e c t r i c c o u p l i n g s u b - m a t r i c e s

[ k \ „ I0 = \ [ C m \ T [ P N \ [ B 0 ] d A = [ k ] l n (2.57)A

[* W = l [ C b ]T [ P M ] [ B 0 ] d A = [ k \ l ¥ (2.58)A

P i e z o e l e c t r i c c a p a c i t a n c e s u b - m a t r i x

{ k \ o = \ [ B 0 \ T [ P 0 \ d A (2.59)A

S h e a r s t i f f n e s s s u b - m a t r i c e s

[ k s ]b = $ [ C } b ]T [ A s }[C. r b ] d A (2.60)A

\ .k s \b i/ / = \ [ C y b i r \ .As \ [ C y ¥ ] d A = \ k s (2.61)A

[ k s V = J [ C w ]r [As. ][Cm lcM (2.62)A

F i r s t - o r d e r n o n l i n e a r s t i f f l e s s m a t r i x

The following nonlinear element sub-matrices are first order functions of the element

transverse displacement through [ 6 \

1[ k x \ b v = ~ \ i C v b ]T { 6 } T [ B ] [ C b ] d A = [ k x ] T¥b (2.63)“ A

I k i h , , , = ^ \ [ C ¥ b ]T [ d } T { A ] [ C m ] d A = [ k l ]Tmh (2.64)

A

1

A

[*ilr = ^ f i C b ] T [ B ] [ e ] [ C W9, \ d A + J [ C ^ ] r [0]r [fi][C6kM (2.65)“ A A

[*1 W = ]Â = [* ,]«* (2 .66)A


33

F i r s t - o r d e r p i e z o e l e c t r i c c o u p l i n g m a t r i c e s


transverse displacement through geometric matrix [ 6 \

[ * i W = ^ J [ c V J r [ 0 ] 7> / v ] [ S 0 ] < M = [ * , & ( 2 - 6 7 )

[ t i U = { p N \ { B 0 \ d A =[A,1 r<P¥ (2 .68)


electrical displacement through piezoelectric resultant force {Af<j}.

= - \ \ [ C v h]T [N0 }[Cv/b)dA

C T T[ * l i V 0 — — T j [ A 0 \[CVV \clA —~ A

t W V = - ^ \ { C v v \T [ N0 ][C¥ ¥ ]dA

( 2 . 6 9 )

( 2 . 7 0 )

( 2 . 7 1 )

where the force resultant matrix [A is found from the force resultant vector (Ar) as

follows:

N xN v , M = ( 2 . 7 2 )

First-order nonlinear stiffness m atrix due to { N , „ }

The following sub-matrices are first-order nonlinear functions of plate membrane

displacement {vv„,} through the membrane resultant force {A/„, }=[/!]{

1 * 1 Atm = ^ l i C ¥ b \ T i N m \[C„f b \dA ( 2 . 7 3 )

\btff — 7 f lCi /rbl [A * ] [C W \dA — [ A r i^ ~

l A / „ , 1 1 \dA

( 2 . 7 4 )

( 2 . 7 5 )

F i r s t - o r d e r n o n l i n e a r s t i f f n e s s m a t r i x d u e t o { N b }

The following sub-matrices are first-order nonlinear functions of plate rotational

displacement { \ p \ through the membrane resultant force {A/b }=[#]{ k ]


34

l*i,v* h = ^ - J i c vb lT I N b ] [C vb \dA~ A

1*1 Nb h<y = \T [ N b \ [ C VV I dA = \ ^ b“ A

I k \Nb V = | j t C W ? l N b II'C W 1 ^

(2.76)

(2.77)

(2.78)

S e c o n c l - o r d e r n o n l i n e a r s t i f f n e s s m a t r i x

The following nonlinear element sub-matrices are second-order functions of the element

transverse displacement through geometric matrix [ 6 \

1*2 U = - \ [ C ¥ b ]r [ 0 \ T [ A \ [ e \ [ C y b \dA

1*2 W via = [ k 2 \ lb~ A

1*2 V = \T \ e \ T [ A w e w C y y VIA

(2.79)

(2.80)

(2.81)

2.9.3 Element Mass Matrices

Element mass matrices are determined using the variation of kinetic energy term

in the generalized Hamilton’s principle. Using MIN3 element interpolation function, the

kinetic energy variation can be written as

s r = { p h { { S » - m }r ( { H „ }LH „ J { .v „ , } + { H v } |W „ })A

+ k&vb f iH w }+ W f {h wy/ })( LH w ]{Swb }+ L" w y ^ } % iAEquation (2.82) can be written in matrix form as:

(2.82)

S T =

SwbT

M b I ' » V 0 O'S i y I I™]#, [ m ] v 0 0 V

S wm 9 0 0 [ '” 1 m 0 w ,n5 w 0 0 0 0 0 <v0

(2.83)

where the element mass sub-matrices are defined as:

[ m ] b = J l w „ . /a

I '« W = j l " W J

(2.84)

(2.85)a


35

Imiy = JL«,v.„J7' ^ !j 2-L« ,,« ,> >

D,

(2.86)a

(2.87)[m ]„ , = J ( L ff„ J + I H „ J ) r - U 5 . ( LH u ] + l H , ] ) d A

<4 “

2.9.4 Element Aerodynamic and Force Matrices

The element aerodynamic stiffness and aerodynamic damping matrices, and load

vectors are determined by examining the variation of work done by externally applied

loads and electrical charge. The external work variation can be written as:

SW ext = J ({^ V'Z> } T i H u-}+ {H wifr })(AP</ + P U , y , t ) ) d A -

(2 .88 )J Y { P c s )d A

where p ( x , y , t ) is the acoustic pressure loading that could affect the panel in addition to

the supersonic aerodynamic pressure loading. Substituting for the aerodynamic pressure

loading using the first-order piston theory, equation (2.38), the external work variation

can be written in a matrix form as follows:

s w ^ = \

Swb t f

8y/ XSwm

\-a a 0 0 * w b

[c*a lysb \.a a ^l// 0 0 V

0 0 0 0 w m0 0 0 0 WP

Sa_CD,,

[j> \bt/f 0 o' w b Pb\

[.? 1 if/b [S 1 y/ 0 0 V • -1- - Plf/►

0 0 0 0 00 0 0 0 _ LP 0 /

(2.89)

where the element aerodynamic stiffness sub-matrices are defined as:

\ b = { a \ J f w J (cos a \ _H w J, v + sin a [ H w ], v ) d A A

\.a a 1b !]/ ~ \ c i [ H lt, (COS DC \_H _), x ^ in CX [_ // wiff _|» v )

A

[« a \ y b = J a\_H wl// J (cos a \_H w J ,x + sin a [ H w J , v ) dAA

[ a a V = J H W v J (cos a \_H w w J,x + sin a | H u. v J, v, ) clA

(2.90)

(2.91)

(2.92)

(2.93)


36

The element aerodynamic damping matrices are the same as element bending mass

matrices given by equation (2.84) through (2.86)

[ § ~ \ . m \ b ' \ -S \b i f / \ .§ \ y / = (2.94)

Finally, the element load vectors due to acoustic random pressure and applied electrical

charge are defined as:

\ P b }= J [ Hj p ( x , \ \ t ) d AA

(2.95)

\ p v } = \ [H p ( x * y , t ) d AA

(2.96)

W }= “ J K j }dA (2.97)A

For the case of panel flutter analysis and suppression, the acoustic pressure. p ( x , v , t ), will

be set to zero. However, some of the analysis cases given in Chapter 4 will consider the

combined effect of acoustic loading and supersonic aerodynamic pressure.

2.9.5 Element Equations of Motion

Using the element stiffness, mass, aerodynamic matrices and load vectors, the

fully coupled nonlinear electromechanical equations of motion for a composite laminate

with piezoelectric layers subject to supersonic aerodynamic and acoustic loading using

the MIN3 element can be written as:

’ ["*k [ m ] h v 0 0" ii'b ~ [ g \ b [«? \btf/ 0 0‘[ m \ v 0 0 ¥ ■ + — ti> \if/b 0 0 ¥

0 0 0 V o 0 0 0 0 vvm0 0 0 0 0 0 0 0 w 0

(1-®a ^blf/ 0 o ' ' 0 0 0 0 1

A \-a a ]tf/b 0

t aci 0

0

0

0

0+

0

0^l//0 +

V0 0 0 0 0 [ k ] 0 _

~ [ k s ]b \b y / 0 o ' 0 \bi f / [*•'1 h r n t ^ ' l I b 0

cc j }y/b [^•5 0 0 [ * I [ * i V t ^ - l ]y/ni [ ^•1 \y /0

0 0 0 0 [*1 h n b [ ^ 1 \m ij/ 0 0

0 0 0 0 _ [ ^ 1 \<pb [ 1 \0 \jf 0 0


37

N 0 f^"IN<p ^bif/ 0 o ' 0 0 '

\.^ \N 0 ^ ip b 0 0+ 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

[ 'lA! b h t 'lA'b h i / / 0 o ' [ •2 [^2 W 0 0 w b P b[ k \ N b \ y b [ k [ Nb V 0 0 + [^2 ]y/f> [*2V 0 0 ¥ __ Ptft

0 0 0 0 0 0 0 0 w m 00 0 0 0 0 0 0 0 / P q

One important thing to note for this equation is that, all element matrices are symmetric

except the aerodynamic stiffness matrices, which are skew symmetric. For abbreviation

from now on, the element bending DOF {vry,} and { y / \ are combined together in a single

vector. Thus, the 4 matrices with subscripts b , b y / , y /b , and y/a re all combined in a single

matrix denoted with the subscript b .

2.10 System Equations of Motion

The system equation of motion for the complete plate can be found by following

the standard finite element assembly, and using the specified structural and electrical

boundary conditions. The electrical DOF also follows the traditional finite element

assembly procedure where the electrical boundary condition stipulates equal potential

across element boundaries for each continuous piezoelectric patch. Using the new

notation for bending DOF:

f o } = 2 b } (2.99)all elenints *■ '

where the summation sign represents the finite element assembly procedure. Combining

the shear and linear bending stiffness matrices in a single stiffness matrix [AH, then the

system equations of motion for a composite panel embedded with piezoelectric layers and

subject to supersonic aerodynamic pressure and acoustic pressure can be written as:

\ M b 0 0" g ~Ga 0 o ' W b '0 M m 0 ■ W m 0 0 0 W m - +0 0 0

%u 0 0 0 0 %

/ "A* 0 o'~ Kb ^bni

1 1£ < 0 o '

A 0 0 0 + Knib Km K/)10 + 0 0 0V

0 0 0 1 * % K<pm 0 0 0


38

K \ b + K \ Nm + K 1NbK lK \

mbpb

K l b m00

ATIbp00

K 2 0 0 0 0 0 0 0 0

w b Pb- = ■ 0

Pp(2 . 100)

where [A/], [ G aJ, [ A a ], and [AT] are the system mass, aerodynamic damping, aerodynamic

stiffness and linear stiffness matrices, respectively; [ATI] and [ K 2 ] are the first-order and

second-order nonlinear stiffness matrices, which are linear and quadratic functions of the

unknown displacements { W b } and { W , „ } respectively; and [ K 1 n 0] is the piezoelectric first

order stiffness matrix which depends linearly on the electrical DOF { W 0 }. In the absence

of acoustic pressure loading, { P b } = 0, equation (2.100) reduces to the problem of

nonlinear panel flutter under yawed supersonic flow. However, by setting A and g a to

zero, equation (2 . 100) describes nonlinear random response of composite panels

subjected to high acoustic excitations. The formulation is kept in a general form with

both effects to allow the study of nonlinear panel response under combined acoustic and

aerodynamic loading. However, {P/,} will be dropped when nonlinear panel flutter

analysis and control are considered.

2.11 Piezoelectric Actuator and Sensor Equations

The system equation (2.100) represents a system of coupled electrical-structural

nonlinear equations. Collecting the structure degrees of freedom {W b ) and {W m ] in one

vector denoted as {Wj, equation (2 . 100) can then be simplified as follows:

( 2 . 101)

M 00 0

K 2 0

fA

'A O'+ ~ K W

r

+V 0 0 _%V i

Av0 ) | / 0 J

Any piezoelectric layer in the composite laminate could be used as a sensor, actuator, or a

self-sensing actuator. First, the case of a plate equipped with piezoelectric sensors and

actuators is examined, then the case of self-sensing piezoelectric actuators.

2.11.1 Piezoelectric Material as Actuators and Sensors

For this case it is assumed that each piezoelectric patch is either used as a sensor

or as actuator. The electrical DOF are partitioned into sensor DOF and actuator DOF, or a

sensor voltage vector and an actuator voltage vector as follows:


39

(2. 102)

Similarly, the piezoelectric coupling and capacitance matrices can be partitioned into

sensor and actuator contributions as given by equation (2.103)

0

K(2.103)

For the piezoelectric sensors, there is no externally applied charge to the sensor

electrodes and hence the electrical loading vector can be written as:

r o(2.104)

Based on the above partition, we can now write the commonly known piezoelectric

sensor and actuator equations.

S e n s o r E q u a t i o n :

From the second row of system coupled equation (2.101) and using equations (2.102)

through (2.104), the piezoelectric sensing equation is:

K 1 }= - f a F ' ( J+ 1 * 1 ] ){l4,> < 2 i 0 5 )

This provides the equation for piezoelectric sensor voltage output. If instead, the sensor is

used as a charge sensor, the sensor output can be determined based on the electrical

relation: q - C V where V is voltage, C the electrical capacitance, and q the electrical

charge. Noting that [K\>] is the piezoelectric capacitance, then the output of a charge

sensor is given by:

{ l ‘ }= - f e . 1+ f c l j J V } (2 - >06)

where { q * } is the collected sensor charge.

A c t u a t o r E q u a t i o n

For the actuator DOF, the second row of equation (2.101) gives:

•([* > ,]+ [k ; }= £ > /] (2.107)

A piezoelectric actuator could be driven either by voltage or charge. For a charge driven

actuator, ] is known and equation (2.107) can be used to find the actuator electrical


40

DOF { w ° ] as a function of the structural DOF {W } and the known applied charge as

follows:

K ° 1= f a J"1 { f a } - ( f a , ]+ f a * . ] M (2- io s )The actuator electrical DOF is then substituted in the first row of equation (2.101) to

obtain the actuator equation for a charge driven actuator. However, in the usual actuator

application, a driving voltage is used instead of a charge. In this case, the actuator

voltage ] is known and the charge drawn onto the piezoelectric electrodes in response

to this voltage is not o f interest. Therefore, equation (2.107) is ignored. For the voltage

driven piezoelectric actuator, the actuator equation can be written as follows:

—-^-[M }+ -^ -[G ]^V J+ (A [A ]+ [^ ll.]+[^'l]+[AT2 ] ){W }= { P } + { P a c l }(2.109)c o - u 0

where {Facr} represents the piezoelectric force due to the voltage at both sensor and

actuator and is given by:

{P a c , > = - ( f a . * ] + f a t * ] ) f a M f a * ] + f a t * ] ) K “ J < 2 -1 10 >

Since the actuation voltage is much bigger than the sensor voltage, the sensor term in

equation (2 . 110) is neglected and the actuator equation for a voltage driven actuator is

finally given as:

- y [ M ] }+ [ G ] }+ ( A [ A ] + [ K w ]+ [ K 1]+M o 01 o (2 .111)

l K 2 ] ) { 1 V } = { P } - ( f a * ] + f a i t * ] ) f a }

2.11.2 Piezoelectric Material as Self-Sensing Actuators

For these devices, each layer of piezoelectric material is used as sensor and

actuator simultaneously. Therefore, there is no need to partition {W 0 ] for this case.

Equation (2.101) can be then written as two separate equations to represent actuation and

sensing as follows:

—V [ M ] } + [ G ] f y } + ( A [ A ] + [ K w ]+ [ATI]co ~ M a (2 . 112)

+ [JC2 ] ) { W }= {/>}- ( fa „ * ]+ f a l ] ) f a ]


41

(2.113)

The key to self-sensing piezoelectric actuators is measuring the electrical charge drawn to

the electrodes, ( P 0} . This electrical charge can then be combined with the piezoelectric

capacitance and applied electrical voltage, the term [ K P] { W 0 ) in equation (2.113), to get a

signal proportional to the structure DOF. Analog circuits such as the ones given in [80]

and [81] can be used to implement the right hand side of equation (2.113). In practice, the

compensation for the piezoelectric capacitance is not perfect and could affect the sensor

performance. In this study, the difference between the piezoelectric capacitance and the

electric circuit compensation will be neglected, i.e., perfect implementation of the right

hand side of equation (2.113) is assumed. Therefore, the sensor output charge for self

sensing piezoelectric actuators is given by:

Thus, equations (2.112) and (2.114) are the actuator and sensor equations, respectively,

for self-sensing piezoelectric actuators.

(2.114)


42

Acoustic pressure

Yawedsupersonicflow Bonded or embedded

Piezoelectric layers

Composite laminate

Figure 2 .1 Composite panel under yawed supersonic flow and acoustic pressure loading


43

Zn-t-1

ZlZ2

{W*} = {VU..., Vk Vnp}

i

/ "

i1

\\jr

np

vk

x-►

V,

Piezoelectric layers

Figure 2.2 Composite laminate composed of n layers with n p piezoelectric layers


44

(x2, y2)

(x l , y l ) { i» ^2,^3} = {Ai, A2, A3}/A A = ZAj

Figure 2.3 MIN3 element geometry and area coordinates


45

CHAPTER III

MODAL REDUCTION AND SOLUTION PROCEDURE

3.1 Introduction

The nonlinear dynamic equations derived in the pervious chapter represent the

equation of motion in the structural node DOF. These equations could be used to solve

nonlinear panel response for flutter and combined aerodynamic and acoustic loading.

However, it is not efficient to do so due to the large number of structure node DOF and

due to the nonlinear stiffness matrices, which have to be updated. Additionally, a reduced

order system that only retains the important modes of the system should be used for

control system design.

In this chapter, the system governing equations are transformed into the modal

coordinates using the panel linear vibration modes to obtain a set of nonlinear dynamic

modal equations o f lesser number. The reduced modal equations can be easily used to

solve the problems of linear and nonlinear flutter boundaries and to analyze panel

response under combined acoustic and aerodynamic pressures. Linear and nonlinear

panel flutter problems can be solved using either time domain methods such as numerical

integration, or frequency domain methods (eigenvalue problem solution). However, the

problem o f nonlinear panel response under combined aerodynamic and acoustic loading

must be solved using time domain methods. The reduced modal equations o f motion will

also be used to design control laws and to simulate panel flutter suppression. The

following are the main assumptions used in the modal reduction and solution procedure:

• General composite laminate, symmetric or unsymmetric.

• Inplane inertia is neglected. This is justified as the frequencies of inplane

vibration modes are much higher frequency than the out of plane bending modes.

• Self-sensing piezoelectric actuators are assumed. Therefore, equations (2.112) and

(2.114) represent the coupled actuation and sensing equations, respectively.

• The piezoelectric actuators bonded to the surface or embedded within the

laminate produce only bending moment and not inplane force. In other words, the

piezoelectric resultant force, is zero. This is achieved by having a pair of


46

self-sensing piezoelectric actuators at the top and bottom surfaces of the laminate

and by applying equal and opposite voltage to these two actuators (see Figure

3.1). This can be justified because the main purpose for the piezoelectric actuators

is to suppress nonlinear panel flutter and because it was concluded in pervious

work by [101] and [103] that piezoelectric actuators with bending moment are

much more efficient than inplane tension for nonlinear panel flutter suppression.

3.2 G overning Equations

Recall the system equation of motion (sensor and actuator equations) derived in

chapter 2, equations (2.112) and (2.114):

M b0

0

' K b

K m b

K 2 0 0 0

K bm+

K m J

>1- ) 1 w (I vy m J

W h

W ,+

mS a CO,

~Ga O' { * b \ 1+(/I o'

_ 0 0 K , J 0 0_K l N <p o

o

P b

o+

K \ b + K i Nm + K \ Nb

K 1 mb

+

+ (3.1)

f K b0+

1

5 C-

____

_J \

V 1 > I

0 JK )

'Wu

\ W ,(3.2)

m

By setting the piezoelectric resultant inplane force, {yV<j}, to zero, as was assumed in the

pervious section, then all linear and nonlinear element and system stiffness matrices that

are function of piezoelectric inplane force resultant [A^] vanish. In addition, the element

vector o f electrical DOF, {vv'0}, becomes a scalar for each element since the electrical

DOF at the pair o f top and bottom piezoelectric layers are equal and opposite. Therefore,

the following matrices [ATra()], [ K 1 n 0 \ , and [K l b0] become zero matrices, see equations

(2.57), (2.67), and (2.70). Using this in equations (3.1) and (3.2) and by neglecting the

inplane inertia as mentioned in the pervious section, the equations of motion then

become:


47

~ M b O ' \+8“ ~Ga o 'R L(

A X ,O '

+~ K b K bm

_ 0 0 _ R, °>o 0 0 R, [ \ 0 0 . K nib K m .

A c t u a t o r E q u a t i o n :

11

U o

' K \ b + K \ N m + K \ N b

mb

S e n s o r E q u a t i o n :

+

~ K 2 O' j+

J . 0 0 /' W hw,m

' R. o f

K b 00 K ) <3-3>

W = - k * o fo l ^ lW',„f

(3.4)

It is worth mentioning that the pure transverse displacement sensor signal in equation

(3.4) is obtained by summing the charge outputs from the pair of top and bottom

piezoelectric layers (see Figure 3.1). Equation (3.3) can be partitioned to two separate

equations for bending and inplane DOF. The inplane DOF, {W,„}, can be expressed in

terms of the bending DOF, { W h \ . as:

{ W ,„ }= I" ' ([£„,(, ! + ( « ,„ ; , I) { W b } (3.5)

and the system equations in terms of the bending displacement { W i,} as:

- L . [ M b ] ^ b Y — [ G a 1R H ([*J + [ K n l 1) i w b }= { P b R ] R }(3.6)CO~ W o

where the linear and nonlinear stiffness matrices are given by:

[ K l 1 = ( i [ A a ] + [ K b \ ~ [ K b m \ { K m ]- { [ K m b ]) (3.7)

(3.8)[Kfl /L 1 - - [ K b m 1 I K^mb 1 + ) +

+ [ K 2 \ - [ K i b m \ [ K m ] - l ( [ K m b \ + [ K l m b \)

The system equation of motion in structural DOF, equation (3.6), has two major

drawbacks. First, the element nonlinear stiffness matrices have to be evaluated and the

system nonlinear stiffness [ K ^ l \ is assembled and updated at each iteration or numerical

integration step. Second, the number of structure node DOF {W b) is usually very large

which is not suitable for control laws design and computationally costly for both

frequency domain and time domain solution methods. To overcome these drawbacks, the

system equations are reduced using modal transformation and modal reduction based on

the values of modal participation. This results in a system of coupled nonlinear modal


48

equations with much lesser number of DOF that could be used for control design,

numerical integration, and frequency domain solution.

3.3 Modal Transformation and Reduction

Assuming that the panel deflection can be expressed as a linear combination of

some known function as

K } = i > r (D{0r }=[<!>]{<7} (3.9)r= I

where the number o f retained linear modes n is much smaller than the number of

structure node DOF in bending, {W b ) . The normal mode {$-}, which is normalized with

the maximum component to unity, and the linear natural frequency cor are obtained from

the linear vibration of the system.

c o r 2 [ M b ][<pr ) = [ K b ]{<t>r } (3.10)

Since the nonlinear matrices [ K l mb\, [K l bm], [ K l Nb], and [ K 2 ] are all function of the

unknown bending DOF { W b } , they can be expressed as the sum of products of modal

coordinates and nonlinear modal stiffness matrices as

( [ K l „ l b l [ K \ b m \ [A Tl^]) = ^ q r i K \ m b ]( r \ [ K l b fJ r>, [ K l tWb1 r>)

(3’l l )

r = I j = 1

where the super-indices of those nonlinear modal stiffness matrices denote that they are

assembled from the corresponding element nonlinear stiffness matrices. Those element

nonlinear stiffness matrices are evaluated with the corresponding element components

[ w b } {r ) obtained from the known system linear mode {<j)r }- Therefore, those nonlinear

modal stiffness matrices are constant matrices. This is a great advantage, as these

matrices do not need to be evaluated at the element level and assembled at each iteration

or time integration step.

The first-order nonlinear stiffness matrix [A7m„] is a linear function o f the inplane

displacement { W „ ,} . From equation (3.5), {W m ) consists o f two terms as:


49

Y q r [ K \ m b ] ^ U ] { « } l r = l J (3.12)

n n n

= ' S Cli'{0r}m ~ ^ ^ c l r c l s (0r.v I w r= i r= l 5=1

where the two inplane modes corresponding to the r-th bending mode {<pr } are defined as:

Thus, the nonlinear stiffness matrix [ K I Nm\ can be expressed as the sum of two nonlinear

modal stiffness matrices as:

[* l/v » .]= -2 > < -[K l/v » ,f r) (3.14)r = 1 r = l , v = l

The nonlinear modal stiffness matrices [ K l Nm](r) and [K 2 Nm ](rs> are assembled from the

corresponding element nonlinear stiffness matrices. Those element nonlinear stiffness

matrices are evaluated with the corresponding element components obtained from

the known inplane modes {#-},„ and {0„}m, respectively. Thus, the nonlinear modal

stiffness matrices are constant matrices. Using equations (3.11) and (3.14), the system

equation, (3.6), is transformed to the following reduced nonlinear system in the modal

coordinates

- V [ M b ^ | c ]{<)}+ 2 { r <or [/ ]& }+C0o CO- (3.15)

( Ik l 1+ K ] + [ x m ] ) { q } = & h [ ic b ( , ] { j y „ }

And the sensing equation in the modal coordinates is:

{7 5 } = - [ ^ f e } (3.16)

As for the electrical DOF, there is no need to do reduction since the number of electrical

DOF, { W 0 ) , is usually small.

The modal matrices are given by:

® v 7 j , [ C ] , [ ^ ] ) = [ 0 f ( [ M 6 ],[C ],[ /i:L ])[cI)l (3.17)

and the quadratic and cubic terms in modal coordinates are:


50

K ]= [<*>f £ qr ][Arm r 1 [/nm61‘1+mb r> -[KiNm t ‘r =1 (3.18)

+[K1 Nb 1rl - [K\bm f r) [K m r 11 [Kmb ])[4>]

n n[k w ] = t-f f s s ‘u M " ' r ' -

r= l„ l (3.19)

[Kif,„,]<r» [*:mr ,[ i„,1>]<),)[0]and the modal load vector and piezoelectric control force are:

\ F b }= W {P h } (3.20)

[ ^ ] = t < i - r k j = [ ^ r (3.2i)

The nonlinear first order modal matrix [AT(/] usually represents a softening effect while the

second order nonlinear matrix [ K qq] represents a hard spring or stiffening effect. For the

special case of symmetric panels, [ K q \ is zero (due to [B ] = 0) and the modal equations of

motion reduces to a multi DOF coupled Duffing system. Structural damping modal

matrix, [ c ] = 2 £ r a ) r ( M r / c o ^ ) [ l ] , has been added to the system modal equation (3.15)

where £ r, Q)r, and M r are the modal damping, frequency, and mass, respectively, and [/] is

the identity matrix. The modal damping ratio can be determined based on testing or on

performance of similar structures.

The advantages of the modal equations of motion given by equation (3.15) are: (i)

there is no need to assemble and update the nonlinear stiffness matrices at each iteration

since all the nonlinear modal matrices are constant, and (ii) the number of modal

equations is much smaller than the structure equations. This approach has been

successfully used for nonlinear panel flutter of composite panels at elevated temperature

by Zhou et al. [104]. This approach has also been demonstrated for nonlinear panel flutter

of composite panels under yawed supersonic flow by Abdel-Motagaly et al. [51] and for

nonlinear response of composite panels under combined acoustic and aerodynamic

loading by Abdel-Motagaly et al. [64],

The determination of the number of modes required for a specific problem is not a

trivial task and requires some special attention to make sure the important modes are


51

retained. The few influential modes to be kept can be determined by the modal

participation value, which is defined as:

For the problem o f nonlinear panel flutter, the number o f modes used can be determined

based on accurate and converged magnitude and frequency of limit-cycle oscillation

(LCO). By doing this, it is guaranteed to get accurate flutter mode shape, which is crucial

for panel flutter suppression problem. For the problem of acoustic loading in the form of

uniform random pressure input, the lowest few symmetric modes are usually used. In

summary, modal reduction is problem dependent and should be performed based on a

systematic modal convergence and participation procedure.

3.4.1 Time Domain Method

Nonlinear panel response could be simply solved by using numerical integration

of the modal dynamic equations of motion. For the case of combined loading, the random

surface pressure is generated and the panel response is determined at each time step. For

the case of nonlinear panel flutter, numerical integration is carried out using any arbitrary

initial conditions at a given value of nondimensional dynamic pressure, A . After some

time, the solution converges to a limit cycle and the limit cycle oscillation (LCO)

amplitude and frequency can then be determined at the given A . This approach is accurate

and general as it could easily handle both structural and aerodynamic damping. However,

the frequency domain solution introduced in the next subsection is more effective

computationally for the case of nonlinear panel flutter when the interest is only to

determine the panel LCO response and the flutter mode shape. For control design and

simulation, numerical integration must be used.

3.4.2 Frequency Domain Method

3.4.2.1 Critical Flutter Boundary

Linear panel flutter equations of motion can be determined by neglecting the

nonlinear stiffness matrices, and by setting the piezoelectric actuation and random surface

pressure to zero in equation (3.15).

P a r t i c i p a t i o n o f t h e r - t h m o d e = \ q r \ / Ski (3.22)

3.4 Solution Procedure


52

[ M b M + ^ [C ]{?}+ 2C r O r [/]{?}+ W L ]fe}= {0}CO

(3.23)

This equation represents a dynamic eigenvalue problem and a solution for the unknown

modal coordinates {q ) can be assumed in the form of:

where [ q , , ] is the eigenvector and Q = 0 + i c o is the panel motion parameter (J.3 is

damping rate and co is frequency). Flutter will occur if the panel motion becomes

unstable, i.e., when the damping rate becomes positive. Using the fact that [G] = [M h \ and

neglecting the structural damping, the linear panel flutter eigenvalue problem can be

written as follows:

where the eigenvalue, Kj, is defined in terms of aerodynamic damping and reference

frequency as:

As the nondimensional dynamic pressure, A , increases, the panel symmetric stiffness

matrix will be perturbed by the skew symmetric aerodynamic matrix (see equation (3.7)

for the definition of [Aj.]). This will result in one eigenvalues K) to increase and another

eigenvalue K j to decrease until they coalesce. If A is increased further, the two

eigenvalues become complex conjugate:

For the case of zero aerodynamic damping, g a = 0, the damping rate is zero at the

coalescence of the two eigenvalues and hence this defines the flutter critical boundary,

A cr , (i.e. at Ki - 0). In the presence of aerodynamic damping, the damping rate will vanish

after the coalescence. Substituting (3.26) into (3.27) and using the definition of Q, it is

found that 0 = 0 when Ki/ K r = g a . The value of A cr in the presence of aerodynamic

damping is thus slightly higher than that with no aerodynamic damping. Based on linear

theory, the panel motion will grow exponentially beyond A cr_ In reality, this is not the

case due to inplane stretching forces. Thus, the nonlinear matrices must be considered to

(3.24)

(3.25)

X/.2 — Kr + I K i (3-27)


53

determine the panel limit-cycle oscillation (LCO) response beyond the critical flutter

boundary.

3A.2.2 Nonlinear Flutter Limit-Cycle Oscillation

The reduced nonlinear modal equations for the nonlinear eigenvalue problem of

panel flutter could be solved using the linearized updated mode with nonlinear time

function (LUM/NTF) approximate method introduced and applied by Gray [11] for

nonlinear hypersonic panel flutter and by Xue [39] for nonlinear supersonic panel flutter.

The LUM/NTF method is an iterative method. It was originally applied in the structure

node DOF and thus requires evaluation and assembly of element matrices at each

iteration. An efficient solution procedure, presented here for the first time, is to apply the

LUM/NTF method to the reduced modal equations. The presented solution procedure is

efficient and has a great advantage in computation time and thus could be used for the

study of nonlinear LCO amplitude and for flutter mode shape determination of composite

panels.

The LUM/NTF solution procedure is based on using harmonic response

assumption and on linearization of nonlinear time functions. Since the intent is to

determine LCO amplitude and frequency, the solution of the modal DOF is assumed in a

harmonic form similar to equation (3.25):

{<y}= { q 0 = { q 0 } ( c o s c u t + i s i n c o t ) (3.28)

Using equation (3.15), the nonlinear panel flutter eigenvalue problem can be written as

follows:

( - K j [ M b ]+ \ K L ]+ [ K q ]+ [if w ] ){ ,„ = {0} (3.29)

For a stable limit cycle, the modal solution in equation (3.28) can be written by setting /?

to zero and by choosing either cos(tu t ) or sin(ry t ) for the harmonic solution. Using this

modal solution, equation (3.29) becomes:

(- K j [m b ]cos cut + cos cut + \ fC q ]cos 2 cut + [K q q ]C°S3 O x ) { Q o }= {0} (3.30)

Using linear approximation for the nonlinear time functions (cos2cut and cos3tuf), the

linearized eigenvalue problem for LCO of nonlinear panel flutter can be written as:


54

2 4v y

Equation (3.31) represents a nonlinear eigenvalue problem where the matrices \ K q ] and

[ K q q ] are function of the unknown eigenvector { q 0 }- It can be solved using an iterative

linearized eigenvalue solution procedure. The nonlinear stiffness matrices are evaluated

at iteration j + I using the eigenvector solution from iteration j . The eigenvalue and

eigenvector can then be determined based on certain convergence criterion (see Figure

3.2). For a given maximum panel deflection ( W m a x ) , the LCO occurs at a specific

nondimensional dynamic pressure value denoted as A l c o - At this value of dynamic

pressure the damping rate vanishes (/? = 0). If the aerodynamic damping is neglected, this

will occur when two eigenvalues of equation (3.31) coalesce. In the presence of

aerodynamic damping, the value of A l c o is reached after of Kj become complex conjugate

( k = K r ± iK i ) at the point where g a = * 7 1 4 K R (s im ile to the linear flutter case).

Figure 3.2 describes the search algorithm and the iterative linearized eigenvalue solution

procedure used to determine A ^ c o for a given LCO amplitude.


Sensor chargesoutput, q

Self-sensing piezoelectric actuators

W62= - W4>1

Figure 3.1 Configuration of self-sensing piezoelectric actuators for bending

moment actuation and transverse displacement sensing


56

S t e p I : For a Given LCO amplitude W m a x , Assume {q a}

S t e p 2 : Assume a value for X

S t e p 3 : Solve the linearized eigen equation for the j-th iteration:

K j \ M b ] (< ? o } j = ( [ ^ z . ] + [ ^ N L } y

[K N L . ] = ^ - [ K q f e o } j _ ! ) ] + ^ \K qq } j - [ ) ]

where { q 0 }j is the updated modal solution

S t e p 4 : Get { W h \ from { q „ } } using { W b ] = [ 0 ] ( q „ a n d Find ( W ntax) j

then adjust { q „ ) j to result in the required W max using:

r -i r i ( ^ m a x ) g iv e nh o i i - h o i j - M — T—

v r max / j

S t e p 5 : Test for convergence of {r/„}, if not go to step 3

S t e p 6 : Check for coalescence of the eigenvalues:

- if not, A = A + A A then go to step 2

- if yes A l c o = A then exit

Figure 3.2 Iterative solution procedure for nonlinear panel flutter limit-cycle response


57

CHAPTER IV

FINITE ELEMENT ANALYSIS RESULTS

4.1 Introduction

Validation of the MIN3 finite element modal formulation and analysis results are

presented in this chapter. The MIN3 finite element modal formulation is validated by

comparison with other Finite element and analytical solutions. The validation tests

performed include: linear free vibration, piezoelectric static actuation, linear and

nonlinear panel flutter, time and frequency domain solutions, and nonlinear panel

response under random acoustic pressure loading.

Using the verified MIN3 Modal formulation, analysis results for the effect of

arbitrary flow yaw angle on nonlinear supersonic panel flutter for isotropic and composite

panels are presented using the frequency domain solution method. The effect of

combined supersonic aerodynamic and acoustic pressure loading on the nonlinear

dynamic response of isotropic and composite panels is also considered.

Results are presented for isotropic material (Aluminum) and for graphite/epoxy

Fiber reinforced composite with different laminate stacking sequence. The properties of

the different materials used in this chapter are given in Table 4.1. As for the panel

geometry, the MIN3 triangular element is used to model square, rectangular, and

triangular panels using different mesh sizes. Figure 4.1 shows a typical MIN3 finite

element mesh used to model both rectangular and triangular panels.

4.2 Finite Element Validation

4.2.1 Natural Frequencies

The First validation test is comparing the First few natural frequencies versus

analytical values for a simply supported square isotropic panel performed to verify the

linear stiffness and mass matrices. The panel dimensions are 12x12x0.05 in.,

corresponding to panel length, a , panel width, b , and panel thickness, h , respectively. The

analytical natural frequencies (rad/s) are determined using the classical plate theory

solution and are given by:


where D = -— . Table 4.2 show the first six natural frequencies calculated using12(1- 1/ - )

equation (4.1) and a 12x12 MIN3 Finite element mesh (total of 288 elements). It can be

seen that the difference between the MIN3 solution and analytical solution is less than

0.5%.

4.2.2 Piezoelectric Static Actuation

The piezoelectric finite element formulation is validated using the cantilevered

bimorph beam shown in Figure 4.2 (symmetric half o f the beam). The bimorph beam

consists of two identical PVDF piezoelectric layers with polling axis in the 3 direction

but with opposite polarities. Hence, this beam will bend when electrical field is applied in

the vertical direction. The mechanical and electrical constants for PVDF are given in

Table 4.1. A unit electrical voltage is applied across the thickness and the beam lateral

deflection at different points is determined. The results obtained using a 5x2 MIN3

elements mesh are compared against the analytical solution of [69], the solid finite

element of [78], and the QUAD4 finite element of [77]. Table 4.3 shows that the

deflections obtained using the MIN3 element formulation are in good agreement with

other analytical and finite element solutions.

4.2.3 Linear and Nonlinear Flutter

The MIN3 modal finite element solution using the lowest 25 modes and a 12x12

mesh is used to compare the critical nondimensional dynamic pressure, A cr, values versus

the CQ conforming quadrilateral element solution [46] and analytical solutions [15] for

simply supported isotropic rectangular panels with different aspect ratios and at different

flow yaw angles. All solutions assume zero aerodynamic damping and A c r is

nondimensional with respect to the panel length, a . Table 4.4 shows that, the solution of

the MIN3 modal formulation is in very good agreement with the CQ element and with

the analytical solution for different aspect ratios and different flow angles with maximum

difference of less than 2%.


59

The limit-cycie oscillations (LCO) response for a simply supported (with

immovable inplane edges) square (12x12x0.05 in.) isotropic panel with zero yaw angle

using a 12x12 MIN3 mesh and 6 modes in the x direction are shown in Figure 4.3.

Results using both frequency domain and time domain solution methods are shown in

this Figure. The LCO obtained by [15] with a 6-mode model using G alerkin 's method and

numerical integration is also shown in this figure for comparison. The aerodynamic

damping coefficient, c a , is set to 0.1 (see equation 2.37). Both time and frequency domain

solutions using the modal MIN3 element formulation are shown to be in good agreement

with the analytical solution.

For composite panels, the limit-cycle results of a simply supported 8-layer [0/45/-

45/90]s graphite/epoxy square (12x12x0.048 in.) panel obtained by [40] using the C 1

conforming rectangular plate element and time domain-modal formulation are compared

with those obtained using the present formulation with time domain solution. The

complete plate is modeled with 12x12 mesh and using 6 modes in x direction. The LCO

response at zero yaw angle, shown in Figure 4.4, demonstrates the accuracy of the MIN3

element and the present Finite element formulation.

4.2.4 Nonlinear Random Response

The last validation test is intended to verify the panel nonlinear random vibration

under acoustic pressure loading. Table 4.5 shows the root mean square (RMS) of the

panel maximum deflection divided by panel thickness ( W may/ h ) obtained using present

formulation and other available solution methods for different sound pressure levels,

SPL, (see section 4.4.1 for the details o f the acoustic pressure loading). The MIN3 Finite

element modal formulation is used for a rectangular isotropic panel (15x12x0.04 in.)

using a 12x12 mesh. The lowest four symmetric modes are included for uniform input

random pressure distribution analysis. The Fokker-Planck-Kolmogorov (FPK) equation

method [62] is an exact solution for the single DOF forced DufFing equation. The Finite

element/equivalent linearization (FE/EL) method [63] assumes that the equivalent

linearized system is stationary Gaussian, whereas the present time domain numerical

integration method does not assume that the displacement response is Gaussian.

Therefore, the present method should be more accurate.


6 0

4.3 Effect of Flow Yaw angle on Nonlinear Panel Flutter

The effect of flow yaw angle, a , on nonlinear panel flutter LCO is presented in

this section. The panels considered include: isotropic and composite material, square,

rectangular, and triangular panels. The frequency domain solution method is used in this

section since it is more efficient computationally than the time domain numerical

integration method. The aerodynamic damping coefficient, ca, is set to 0.01 for all cases

and structural aerodynamic damping is neglected.

4.3.1 Isotropic Panels

A simply supported square isotropic panel (12x12x0.05 in.) with immovable

inplane edges ( u = v = 0) at different flow angles is studied in detail. First, convergence

of the LCO response using various finite element mesh sizes is studied. Three mesh sizes

o f 8x 8 (128 elements), 10x 10 (200 elements) and 12x 12 (288 elements) are used to

model the simply supported square isotropic panel. Limit-cycle amplitude values versus

nondimensional dynamic pressure using 16 modes, mode (1,1) to (4,4), for the three

mesh sizes at a = 45° are shown in Figure 4.5. The maximum difference of the

nondimensional dynamic pressure between the 10x10 and 12x12 models in Figure 4.5 is

less than 1 % .

Modal convergence is also studied in detail for the square panel at different flow

angles. The panel is modeled using a 12x12 mesh or 288 MIN3 elements. The number of

structure DOF { W t , } is of 407 and it is reduced to the modal coordinates to include the

selected n modes. The modal participation values using 4 modes in .t direction and 4

modes in y direction (4x4 modes, total of 16 modes) for two limit-cycle amplitudes

W mCLXl h = 0.01 and 0.8 at a - 0°, 45° and 90° are given in Tables 4.6. The modal

participation values indicate that a 4-mode model in the flow direction would yield

accurate and convergent limit-cycle results for a = 0 and 90°, while a combination of .r

direction and y direction modes is required for a = 45°. The modal convergence is further

confirmed by the convergence o f the nondimensional dynamic pressure versus the

number of modes used in model reduction as shown in Figure 4.6 for 0° yaw angle and in

Figure 4.7 for 45° yaw angle. These figures confirm that, a 16 modes model is required

for all yaw angles to yield convergent LCO amplitude. For the 0° yaw angle case, Figure


61

4.6 shows that using the well-known 6 modes in x direction and using 16 modes, (1,1) to

(4,4), yield almost the same LCO amplitude. From symmetry, the same will apply for the

90° yaw angle case. The 16 modes model is used in order to have one model for all yaw

angle that can be used for the control system design in the next chapters.

The variation of limit-cycle amplitude versus nondimensional dynamic pressure

for a = 0°, 15°, 30°, and 45° are given in Figure 4.8. It is seen that increasing the flow

angle has the effect of slightly increasing the critical nondimensional dynamic pressure,

A cr , at zero limit-cycle amplitude (linear flutter boundary) for this square isotropic panel.

However, as the LCO amplitude, W niax/ h , increases the difference in X between different

flow angles decreases at fixed W maJ h . This effect is due to the different nonlinear effect

resulting from having different flutter mode shapes as the flow angle changes. Figure 4.9

shows the effect o f aspect ratio (a / b ) on the nondimensional dynamic pressure at a = 0

and 45°. It is seen that increasing the flow angle for isotropic panels with ( a / b ) > 1 causes

the nondimensional dynamic pressure to increase at fixed limit-cycle amplitude.

The flutter deflection shapes at W mcLX/ h = 1.0 for flow angles of a = 0°, 45°, and

90° are given in Figure 4.10. It is clear how the flow angle greatly changes the flutter

deflection shape. This necessitates having different controllers and different actuator and

sensor locations for different flow angles as will be studied in detail in Chapter 6 .

4.3.2 Composite Panels

Two simply supported rectangular (15x12x0.048 in.) graphite/epoxy composite

panels with two different laminate stacking sequences are considered. Both panels are

modeled using a 12x12 MIN3 finite element mesh. The first laminate considered is an 8-

layer [0/45/-45/90]s panel. Table 4.7 shows the modal participation values for this panel

using the lowest 16 modes in increasing frequency order. Figure 4.11 shows the LCO

amplitude convergence using different number o f modes and confirms that convergent

LCO amplitude can be obtained using the 16 modes model for this panel. Figure 4.12

shows the effect o f flow yaw angle on the nonlinear panel flutter response and Figure

4.13 shows the same effect on flutter deflection shape. Figure 4.12 shows that as a

increases from 0 to 90°, A increases at the same LCO amplitude, W maxf h . However, as

W ,„ a x /h increases the difference in X decreases due to the different nonlinear effect for the


6 2

different flutter deflection shapes resulting for different flow angles, as shown in Figure

4.13.

The second composite panel considered is a 3-layer [-40/40/-40] laminate. Modal

convergence is given by Table 4.8 and Figure 4.14 which confirm the validity of a 16

modes model for this panel. Figures 4.15 and 4.16 show the effect o f flow yaw angle on

nonlinear panel flutter response and on flutter deflection shape, respectively. It is seen

how the nondimensional dynamic pressure, A , at 45° yaw angle is less than that of 0° and

90° at the same limit-cycle amplitude. The laminate considered is a good example to

demonstrate the importance of the effect of the flow yaw angle on the nonlinear panel

flutter. It is also seen that, unlike the isotropic case, the flutter deflection shape for a — 0°

and 90° is clearly not symmetric about x or y. This is because the [-40/40/-40] composite

laminate is anisotropic.

4.3.3 Triangular Panel

To take advantage of the triangular MIN3 element, a clamped isotropic triangular

panel is also considered. The panel dimensions are 12x12x0.05 in. and the panel is

modeled using a 10x10 Finite element mesh (see Figure 4.1). Figure 4.17 shows modal

convergence for this panel using the lowest 16, 25 and 36 modes. Based on this Figure a

25 modes model is used for this panel. Figure 4.18 shows the effect o f flow yaw angle on

nonlinear panel response. It is noted that the linear flutter boundary for <2=0° and 180° is

the same, but A becomes different as the LCO amplitude increases because of the

different nonlinear effect for the different flutter deflection shape of 0° and 180° yaw

angles. The same applies for a = 45° and 225°. It is also seen that the worst case for this

panel is when the flow is yawed with 315° (^-5°), i.e., along the tilted edge.

4.4 Effect of Combined Supersonic Flow and Acoustic Pressure Loading

4.4.1 Random Surface Pressure

The input acoustic surface pressure is assumed as a band limited Gaussian random

white noise that is uniformly distributed over the panel surface. The pressure time

sequence is generated by using a randomly generated Gaussian pressure Filtered by a

Chebyshev low pass Filter with the speciFied cut-off frequency, f c . The pressure power

spectral density (PDS) is given by:


63

S ( . f ) = P o i q y 0 < / < f c(4.2)

= 0 f > f c

where p a is the reference pressure (p a = 2 .9xl0 '9 psi) and SPL is the sound pressure

spectrum level measured in decibels. The power spectral density (PSD) of the input

pressure at SPL = 90 dB is shown in Figure 4.19. In practice, the acoustic pressure

loading could be determined from recorded flight data. The formulation derived in

Chapters 2 and 3 combined with time domain numerical integration solution is general in

the sense that it could handle both stationary Gaussian as well as non-stationary non-

Gaussian random loads.

4.4.2 Isotropic Panel

The response of a simply supported square isotropic panel (12x12x0.05 in.)

under combined aerodynamic and acoustic pressure loading is studied in detail. The panel

is modeled using a 12x12 MIN3 element mesh. The modes considered are (1,1) to (6,1)

for panel flutter and (1,1), (1,3), (3,1), and (3,3) for random uniform pressure loading. No

modal participation values are needed for this well studied problem. The aerodynamic

damping coefficients, c a , is assumed to be 0.01 and a structural modal damping of 1% is

added to all modes. No flow yaw angle is introduced for this analysis.

Figure 4.20 shows the root mean square (RMS) of panel maximum deflection

divided by panel thickness for SPL = 0, 100, 110, and 120 dB. The case of 0 dB SPL

corresponds to the case of nonlinear panel flutter and the points at A = 0 correspond to

conventional nonlinear panel response under acoustic excitation only. The maximum

deflection is located at three quarter length from the leading edge (3a/4, a/2) for panel

flutter and at plate center (a/2, a/2) for pure acoustic excitation. The maximum location

for the combined load cases is somewhere between 3a/4 and a / 2 depending on the values

of SPL and A . It is seen from Figure 4.20 that a stiffening effect causes the RMS of

maximum deflection to decrease as A increases until flutter A cr is approached. This is due

to the anti-symmetric aerodynamic stiffness matrix perturbation, which increases the

frequency of mode (1,1) and decreases the frequency of mode (2,1). Mode (1,1) is a

major contributor mode to the panel random response while the anti-symmetric mode

(2,1) has no effect on the panel random response. The RMS of maximum deflection starts


64

increasing again as soon as k cr is reached. This figure clearly demonstrates the

importance o f the effect of combined aerodynamic and acoustic loading on the panel

nonlinear response.

Time and frequency response of the panel maximum deflection is studied in detail

for five different combinations of SPL and k . The results are given in Figures 4.21

through 4.25. Figures 4.21 and 4.22 represent the panel response at low (100 dB) and

high (120 dB) SPL, respectively. At low SPL, the panel experiences small deflection

within the linear random vibration with mode (1,1) being the dominant mode. At high

SPL, the panel experiences large deflection nonlinear random vibration. The nonlinear

behavior at SPL = 120 dB is demonstrated in Figure 4.22 via the broadening and shifting

to higher values of the frequency peaks in the deflection PSD. At k = 800 and 0 dB SPL,

the panel response is the expected pure panel flutter limit-cycle oscillation with a single

dominant frequency, as shown in Figure 4.23. For the combined load case of k — 800 and

SPL = 100, shown in Figures 4.24, the panel response is dominated by the panel flutter

behavior as the random pressure effect is small compared to the aerodynamic loading.

This is clear from the single dominant frequency in the deflection PSD plot. For the case

of k = 800 and SPL = 120, shown in Figure 4.25, the panel response is dominated more

by random vibration since the SPL is high. This is demonstrated in the PSD plot by the

broadening of the frequency peaks.

4.4.3 Composite Panel

The effect of combined loading for a clamped rectangular (15x12x0.048 in.) 8-

layer [0/45/-45/90]s graphite/epoxy panel is also considered. The panel is modeled using

a 12x12 MIN3 Finite element mesh. The system equation of motion is reduced using the

lowest n linear vibration modes. Table 4.9 shows the effect o f the number of modes used

on the RMS of panel maximum deflection at k - 800 and SPL = 120 dB. It is shown that

a 16, 20, or 25 modes model would yield convergent results. The modal convergence is

further verified by the modal participation of the lowest 25 modes given in Table 4.10.

By retaining the 13 modes with participation values > 1 % in the analysis, a very accurate

RMS o f W max/ h can also be obtained, as shown in Table 4.9. Figure 4.26 shows the panel

nonlinear response under combined loading for different values of k and SPL. These


65

results are obtained by using the most dominant 13 modes defined by modal participation

>1%. Similar to the isotropic case, this figure shows clearly the importance of

considering both aerodynamic and acoustic loading in the analysis o f composite panels

especially at high SPL.

4.5 Summary

Validation of the MIN3 modal finite element formulation and analysis results for

nonlinear panel flutter and for nonlinear panel response under combined aerodynamic

and acoustic pressures are presented in this chapter. The finite element formulation is

validated by comparison with other finite element and analytical solutions including

validation of linear free vibration frequencies, piezoelectric static actuation, linear and

nonlinear panel flutter, and nonlinear panel response under random acoustic pressure

loading. Analysis results for the effect o f arbitrary flow yaw angle on nonlinear

supersonic panel flutter for both isotropic and composite panels are presented using the

modal LUM/NTF solution method. Results are presented for square, rectangular, and

triangular panels. The results showed that the effect of flow yaw angle completely

changes the shape of the panel limit-cycle deflection. It also showed the effect of the yaw

angle to be a very important parameter especially for composite panels where the flow

direction may increase or decrease the nondimensional dynamic pressure at fixed limit-

cycle amplitude depending on the panel lamination. The effect of combined supersonic

aerodynamic and acoustic pressures loading on the nonlinear dynamic response of

isotropic and composite panels is also presented. It is found that for panels at supersonic

flow, only acoustic pressure (sonic fatigue) is to be considered for low dynamic pressures

( A « A c r ) while both acoustic and aerodynamic pressures have to be considered for

significant and high aerodynamic pressures.


6 6

Table 4.1 Material properties for the different materials used for

finite element validation and analysis results

Isotropic (aluminum)

E 10.5 Msi

V 0.3

P 0.2588x1 O'3 lb-sec2/in.4

Composite (graphite/epoxy)

H, 22.5 Msi

H: 1.17 Msi

G ,2 0.66 Msi

G23 0.44 Msi

V12 0.22

P 0.1458x10'3 lb-sec2/in.4

PVDF (for bimorph pointer)

E 2.0 Gpa

V12 0.29

d3l = d32 2.2e-l 1 m/V


67

Table 4.2 Comparison between natural frequencies using MIN3 element

and using analytical solution for square isotropic panel

Mode (n , m )Frequency (Hz)

12x12 MIN3 Analytical

( 1, 1) 66.5482 66.4903

(2 , 1) 165.9355 166.2259

(2 ,2 ) 266.8552 265.9614

(3,1) 332.4625 332.4517

(3,2) 430.1436 432.1872

(3,3) 564.7975 565.1679

Table 4.3 Comparison of static deflection for the piezoelectric bimorph

beam (10‘7 m) using different methods (see Figure 4.2)

Position 1 2 3 4 5

Analytical [69] 0.138 0.552 1.242 2.208 3.450

Solid FE [78] 0.138 0.552 1.242 2.208 3.450

QUAD4 [77] 0.14 0.55 1.24 2.21 3.45

MIN3 0.149 0.587 1.289 2.256 3.487


6 8

Table 4.4 Comparison of nondimensional linear flutter boundary using

different methods at different flow angles

a/b MethodNondimensional Critical Dynamic Pressure, A cr

ft II o 0 a = 30° a =45° a =60° a =90°

0.5 CQ [46] 382 213 172 151 135

Analytical [15] 385 - - - 138.7

MIN3 Modal 378 215 177 155 138

1.0 CQ [46] 503 516 523 516 503

Analytical [15] 512.6 - - - 512.6

MIN3 Modal 513 522 527 522 513

2.0 CQ [46] 1081 1206 1388 1703 3056

Analytical [15] 1110 - - - 3080

MIN3 Modal 1117 1237 1414 1712 3020

*AI1 values are obtained for neglected aerodynamic damping

Table 4.5 Comparison of the RMS (W max/h) for a simply supported rectangular

(15x12x0.04 in.) isotropic plate under acoustic pressure loading only

using different methods and mode numbers

SPL (dB) FPK [62]

1 mode

FE/EL [63] Present

4 modesI mode 4 modes

90 0.249 0.238 0.238 0.266

100 0.592 0.532 0.533 0.489

110 1.187 1.030 1.031 1.092

120 2.200 1.902 1.905 2.113


69

Table 4.6 Modal participation values at various limit-cycle amplitudes

and different flow angles for simply supported square panel

Mode

Modal Participation, %

a = 0°

Wmax/h

a = 45°

Wmax/h

a = 90°

Wmax/h

0.01 0.8 0.01 0.8 0.01 0.8

qu 42.18 45.28 31.17 34.41 42.18 45.28

qi2 0.32 0.23 22.62 22.11 42.61 39.17

qi3 0.00 0.79 4.68 4.66 11.95 10.68

q u 0.02 0.03 0.93 1.16 2.44 2.44

q2i 42.61 39.17 22.62 22.11 0.32 0.23

q22 0.25 0.18 9.04 7.38 0.25 0.18

q23 0.00 0.64 1.26 0.77 0.05 0.06

q24 0.01 0.01 0.31 0.30 0.09 0.10

q3i 11.95 10.68 4.68 4.66 0.00 0.79

q32 0.05 0.06 1.26 0.77 0.00 0.64

q33 0.00 0.22 0.09 0.03 0.00 0.22

q34 0.01 0.02 0.01 0.04 0.00 0.08

q4i 2.44 2.44 0.93 1.16 0.02 0.03

q42 0.09 0.10 0.31 0.30 0.01 0.01

q43 0.00 0.08 0.01 0.04 0.01 0.02

q44 0.01 0.02 0.00 0.02 0.01 0.02


70

Table 4.7 Modal participation values at various limit-cycle amplitudes and different flow

angles for simply supported rectangular graphite/epoxy [0/45/-45/90]s panel


Modea == 0° a = 45° a =

oOO'Wmax/h Wmax/h Wmax/h

0.01 0.8 0.01 0.8 0.01 0.8

qi 24.85 29.87 22.74 25.24 25.62 27.43

qa 16.54 12.64 29.59 24.37 41.11 36.57

q3 31.92 33.52 17.47 21.17 3.61 4.04

q-t 5.58 3.45 5.22 4.65 7.19 6.38

qs 1.16 0.07 11.93 9.91 13.13 12.15

q6 15.37 15.03 3.35 5.74 0.12 1.65

q? 0.48 1.01 1.47 0.43 1.55 2.29

qs 0.68 0.21 0.92 0.83 0.28 0.78

q« 0.04 0.07 3.27 3.00 4.08 4.27

qio 0.01 0.18 0.07 0.04 0 .1 1 0.81

qn 3.10 3.38 0.77 1.60 0.01 0.00

qi2 0.02 0.07 0.36 0.01 0.20 0.10

qi3 0.16 0.09 0.39 0.61 0.01 0.01

qi4 0.00 0.06 2.21 2.00 2.89 2.88

qis 0.01 0.09 0.01 0.03 0.01 0.52

qi6 0.00 0.17 0.16 0.30 0.00 0.05


71

Table 4.8 Modal participation values at various limit-cycle amplitudes and different flow

angles for simply supported rectangular graphite/epoxy [-40/40/-40] panel

Mode


a = 0°

Wmax/h

a = 45°

Wmax/h

a = 90°

Wmax/h

0.01 0.8 0 .0 1 0.8 0.01 0.8

qi 25.33 26.38 28.85 28.95 25.65 25.90

qa 41.18 35.26 44.90 40.35 39.10 32.55

q3 4.17 7.01 2.55 2.80 7.94 10.76

q4 11.38 9.79 12.06 11.55 9.87 8.67

qs 2.83 2.73 0.73 1.00 3.45 3.54

qe 6.15 6.75 6.15 7.25 4.52 5.25

q? 1.46 2.54 1.67 2.22 3.08 4.51

qs 2.32 2.13 0.18 0.27 2.10 2.13

qy 0.84 1.22 0.74 1.31 0.32 0.70

qio 1.05 1.55 0.89 1.61 1.06 1.58

qu 1.22 1.21 0.01 0.07 0.95 1.05

qi= 0.13 0.51 0.17 0.67 0.04 0.44

qi3 0.24 0.39 0.20 0.33 0.64 0.97

q u 0.74 1.17 0.65 1.22 0.64 1.01

qi5 0.83 1.02 0.09 0.00 0.42 0.74

q i6 0.07 0.26 0.07 0.32 0.15 0.13


Table 4.9 Modal convergence for clamped rectangular [0/45/-45/90]s

graphite/epoxy panel at SPL = 120 dB and A = 800

Number of Modes RMS( Wmax/h)

1 0.5557

2 0.5845

6 0.7814

9 0.7798

16 0.8279

20 0.8110

25 0.8183

Selected 13 modes 0.8124

Table 4.10 Modal participation for a clamped rectangular [0/45/-45/90]s

graphite/epoxy panel at SPL = 120 dB and A = 800

Mode Participation, % Mode Participation, %

9i 36.72 914 4.19

92 5.24 915 0.38

93 19.3 916 0.99

94 4.25 917 1.38

95 4.01 918 2.55

96 7.67 919 0.21

97 1.71 920 0.27

98 0.76 921 0.53

99 0.33 922 0.33

9io 1.54 923 0.54

9u 4.77 924 0.14

912 0.35 925 1.54

913 0.28


73

ik

b

k --------

Figure 4.1 Typical MIN3 elements mesh used to model rectangular and triangular panels


74

Piezoelectric bi morph beam

I mm

mm0 cm

Clamped end

Figure 4.2 Clamped piezoelectric bimorph beam modeled using MIN3 elements


Wm

ax/h

75

1

0.8

0.6

0.4

0.2 o M IN3 Frequency Domain — M IN3 Time Dom ain A— Analytical

o '—500 550 600 700650 750 800

Nondim ensional Dynamic Pressure

Figure 4.3 Validation of flutter limit-cycle amplitude for simply

supported square isotropic panel


Wm

ax/h

76

0.8

0.6

0.4

0.2 • M IN3 Time Domains Rectangular Time Domain

280 300 320 340 360 380 400 420Nondimensional Dynamic Pressure

Figure 4.4 Validation of flutter limit-cycle amplitude for simply

supported square [0/45/-45/90]s laminate


Wm

ax/h

77

1

0.8

0.6

0.4

0.2 12x 1210x 108 x8

e

o '—500 550 600 650 700 750 800

Nondimensional Dynamic Pressure

Figure 4.5 Finite element mesh convergence for simply supported

isotropic square panel at 45° flow angle and using 16 modes


78

0.8

0.6■s2pm

£0.4

4 m odes 9 m odes 16 m odes 25 m odes6 m odes in x d irection

0.2

300 400 500 600 700 800 900Nondim ensional Dynamic Pressure

Figure 4.6 Modal convergence for simply supported isotropic

square panel at 0° flow angle


Wm

ax/h

79

0.8

0.6

0.4

4 m odes 9 m odes 16 m odes 25 m odes

0.2

400 450 500 550 600 650 700 750 800Nondimensional Dynamic Pressure

Figure 4.7 Modal convergence for simply supported isotropic

square panel at 45° flow angle


Wm

ax/h

8 0

0.8

0.6

0.4

0 degQ 15 deg

—©— 30 deg — 45 deg

0.2

500 550 600 650 700 750 800Nondimensional Dynamic Pressure

Figure 4.8 Effect of flow yaw angle on limit-cycle amplitude for

simply supported isotropic square panel


81

0.8a /h = a /b = 2

0.6

I0.4

0.2

0 deg 45 deg

500 1000 1500 2000 2500Nondim ensional Dynamic Pressure

Figure 4.9 Effect o f panel aspect ratio at different yaw angles on limit-cycle amplitude

for simply supported isotropic panels


8 2

0.5>-

0 0y (in.) x (in.)

?0.5-|

0 0y (in.) x (in.)

a = 45°

?o.sH

0 0y (in.) x (in.)

a = 90°

Figure 4.10 Flutter mode shape at different flow angles for

simply supported isotropic square panel


83

0.8

0.6

0.4

■s— 4 modes -©— 9 modes 16 modes

25 modes

0.2

150 200 250 300 350 400 450Nondimensional Dynamic Pressure

Figure 4.11 Modal convergence for simply supported rectangular

graphite/epoxy [0/45/-45/90]s panel at 45° flow angle


84

0.8

0.6. s*5?cs£

£0.4

0.2 0 deg 45 deg 90 deg

150 200 250 300 350 400 450Nondim ensional Dynamic Pressure

Figure 4.12 Effect of flow yaw angle on limit-cycle amplitude for simply supported

rectangular graphite/epoxy [0/45/-45/90]s panel


85

0 oy (in.) x (in.)

a = 0°

15

'0.5

0 0y (in.) x (in.)

a = 45°

15

0 0y (in.) x (in.)

a = 90°

15

Figure 4.13 Flutter mode shape at different flow angles for simply supported

rectangular graphite/epoxy [0/45/-45/90]s panel


Wm

ax/h

8 6

0.8

0.6

0.4

4 m odes 9 modes 16 m odes 25 modes

0.2

A

100 120 140 160Nondim ensional Dynamic Pressure

180 2 2 02 0 0

Figure 4.14 Modal convergence for simply supported rectangular

graphite/epoxy [-40/40/-40] panel at 45° flow angle


87

0 .8

0.6J5*5?csc

£0.4

0.2 0 deg 45 deg 90 deg


Figure 4.15 Effect of flow yaw angle on limit-cycle amplitude for simply supported

rectangular graphite/epoxy [-40/40/-40] panel


8 8

0.5

0 0y (in.) x (in.)

a = 0°

15

0 0y (in.) x (in.)

1

§rô.5

0

y ( ‘n-) 0 0 x (in.)

Figure 4.16 Flutter mode shape at different flow angles for simply supported

rectangular graphite/epoxy [40/-40/40] panel


89

rA

0 .8

0.6

0.4

0.2 16 m odes 25 modes 36 modesA

3600 3800 4000 4200 4400 4600 4800 5000Nondimensional Dynamic Pressure

Figure 4.17 Modal convergence for simply supported triangular

isotropic panel at 45° flow angle


Wm

ax/h

90

0.8

0.6

0.4

0 deg 180 deg 45 deg 225 deg 315 deg

0.2

30002500 3500 4000 4500 5000Nondimensional Dynamic Pressure

Figure 4.18 Effect of flow yaw angle on limit-cycle amplitude for

simply supported triangular isotropic panel


(psi

)7H

z

-10

-12

500 1000 Frequency (Hz)

1500 2000

Figure 4.19 Power spectral density o f random input pressure at SPL = 90dB


92

1.5

OdB 100 dB H O dB 120 dB

0.5

100 200 300 400 500 600 700 800Nondim ensional Dynamic Pressure

Figure 4.20 RMS of maximum deflection for simply supported square isotropic panel

under combined acoustic and aerodynamic pressures


93

0.6

0.4

0.2

-0.4

- 0.6

- 0.80.1 0.2 0J 0.4

Time (Sec)

20

2-40

-60

-80200 400 600

Frequency (Hz)800 1000 1200

Figure 4.21 Time and frequency response for simply supported square

isotropic panel at SPL = 100 dB and X = 0


>■

0.1 0.2 0_3 0.4Time (Sec)

20

-30

-40

-50 •

-60200 400 800600

Frequency (Hz)1000 1200




-1.51---------------- '---------------- >--------------------------------0.42 0.44 0.46 0.48 0.5

Time (Sec)

40

20

-40

-60

-80

-100500 1000





96

■»ss5£

0.1 03 0.4Time (Sec)

40

-20

> -40

-60

-80500 1000 1500 2000

Frequency (Hz)




97

0.1 0 .2 0 .3Time (Sec)

20

> -40

-60

-80500 1000


igure 4.25 Time and frequency response for simply supported square

isotropic panel at SPL = 120 dB and A. = 800


RM

S(W

max

/h)

98

1.5

0 dB 100 dB 110 dB 120 dB•& r

0.5


1000

Figure 4.26 RMS of maximum deflection for clamped rectangular

graphite/epoxy [0/45/-45/90]s panel


99

CHAPTER V

CONTROL METHODS AND OPTIMAL PLACEMENT OF

PIEZOELECTRIC SENSORS AND ACTUATORS

5.1 Introduction

Many control strategies such as classical, optimal, Ho*, nonlinear, fuzzy logic, and

adaptive control have been utilized for vibration control in the literature. Two major

properties, the nonlinearity and the large dimension of the model, can characterize the

nature o f the problem of nonlinear panel flutter with yawed supersonic flow. Modem

control techniques such as optimal control and H,*, are very suitable for such systems, as

they can easily handle large dimension and no prior knowledge of the controller structure

is required.

In this study, optimal control strategies are the main focus for the suppression of

nonlinear panel flutter. The linear quadratic Gaussian (LQG) control, which combines

both linear quadratic regulator (LQR) optimal feedback and Kalman filter state estimator,

is considered as systematic linear dynamic compensator. To address the issue of the

flutter nonlinear dynamics involved, an extended Kalman filter for nonlinear systems is

also considered and combined with optimal feedback to form a nonlinear dynamic

compensator. Finally, a more practical approach based on optimal output feedback is

used to alleviate the problem of state estimation. Closed loop criteria based on the norm

of feedback control gains for actuators and on the norm of Kalman filter gains for sensors

are used to determine the optimal location of self-sensing piezoelectric actuators.

5.2 State Space Representation

For control design and simulation, the nonlinear modal equations of motion given

in Chapter 3, equations (3.15) and (3.16), need to be cast in the standard state space form.

In state space representation, the system equations o f motion are written in the form of

coupled first order differential equations. State space models provide a standard and

efficient representation of systems with large number o f DOF and of multi-input multi

output (MIMO) systems in the time domain. Linearization about reference equilibrium

points can be used to obtain a linear state space models that could be used for the design


100

of modern optimal and robust control laws. The system states, AT, are defined as the panel

modal deflections and modal velocities:

X = (5.1)

The control input, U, and sensor output, Y, are the self-sensing piezoelectric actuators

input voltage and sensor charge output, respectively:

< / = K ) r = (5.2)

Thus, the system modal equations o f motion in the continuous time domain state space

form are:

X =(A + J A q ) x + BU

Y = CX + DU

where the system matrices are given by:

[0 ] [/]. - ^ K r 1^ ] -coi[Mbr ic d\

[o] [or-col[Mb l {i K q }+[Kqq]j [0]

(5.3)

[ A ] =

U a q ] = (5.4)

[B] =_[0] _ '

- co l [Mb l l [Kb(p] * [ c M k J [0]], [Z>] = [0]

The modal damping matrix [c^ ] contains both modal aerodynamic and modal structural

damping terms. The system matrix [ A A q ] represents the effect of nonlinear stiffness

matrices of the panel. By using linearization about the system reference point (point with

no deflection, {<7} = 0 ), the nonlinear system matrix vanishes and the state space model in

(5.3) reduces to a linear state space model given by:

X = AX + B U Y - C X + D U

If the sensor electrical circuit is set up to measure the time rate of electrical charge

(current) instead of the electrical charge, then the output matrix C is:

[C ]= [[0 ] [ / ^ ] ] (5.6)


101

It has to be noted that for non-ideal self-sensing piezoelectric actuators, i.e., with no

perfect compensation for the piezoelectric capacitance term, the D matrix will be nonzero

and will contain the residual feedforward effect resulting from imperfect compensation.

As discussed in chapter 2, this effect is usually small and hence neglected in this study.

5.3 Control Methods

5.3.1 Linear Quadratic Regulator (LQR)

One of the most commonly used methods to design full state feedback control for

linear systems is the LQR which seeks a solution for the linear full state feedback

problem defined as:

U = - K X (5.7)

that minimizes a quadratic performance index, / , that is function of both system states

and control effort.

J = ° ] [ x T Q X + U T R u ] d t (5.8)0

where Q is a symmetric positive semi-definite state weighting and R is a symmetric

positive definite control effort weighting. The solution for the controller gains that

minimize the performance index of equation (5.8) for the linear state space system give

by equation (5.5) is:

K = R ~ 1 B T P (5.9)

where P is a positive definite symmetric matrix determined from the solution of the

Riccati equation defined as:

P = A T P + P A - P B R ~ 1 B T P + Q (5.10)

In general, the optimal feedback gain sequence is time varying. For linear time invariant

systems, the value o f the optimal gain converges very quickly to a fixed value. The final

fixed value gain could be used as a suboptimal solution since it is easier for

implementation. For the case of fixed gain suboptimal feedback gain, the Riccati equation

reduces to the well know algebraic Riccati equation (ARE):

A 7 P + P A - P B R ~ 1 B T P + Q = 0 (5.11)


1 0 2

The weighting matrices Q and R are used as control design tuning parameters. The

selection of Q and R requires some trials to achieve certain system response

characteristics.

5.3.2 Optimal Linear State Estimation

The application of LQR is not feasible in most cases because it is difficult to have

sensors that measure all system states. In addition, sensor outputs are usually noisy and

the system dynamics is not known exactly. One solution to this problem is the use of

probabilistic approach for both process dynamics and sensor noise. This leads to the well

known Kalman filter state estimator. Kalman filter is a set of recursive mathematical

equations that provide a solution to the least squares optimal state estimation problem in

the presence of process and sensor noise.

Consider a standard linear state space model as the one given in equation (5.5)

with added process and sensor noise:

where rj and v are uncorrelated process and measurement Gaussian white noise with zero

mean and known covariance matrices, Q e = E { r j r f } and R e = E{vvr }. Kalman filter

achieves optimal state estimation by minimizing the covariance matrix o f state estimation

error defined as:

leads to the standard linear Kalman filter estimator described by the following dynamic

equations:

In these equations, the error covariance matrix, Pe, and the estimator gain, Ke, are time

varying quantities. For the special case of linear time invariant systems with time

invariant process and sensor noise, the error covariance time derivative converges very

quickly to zero; hence, the Kalman filter gains become a constant matrix. For this case.

X = A X + B U + n Y = C X + D U + V

(5.12)

(5.13)

where X is the estimated state vector. The minimization of the error covariance matrix

X = A X + B U + K e ( Y - C X )

Pe = A P e + P e A T - P eC T R ; 1CPe + Q e

K e = PeC T R g 1

(5.14)


103

the error covariance matrix is determined using the solution of the estimator algebraic

Riccati equation (ARE) given by:

A P e + P e A T - P e C T R ~ l C P e + Q e = 0 (5.15)

As for the case of LQR, some tuning is required to select R e and Q e in order to achieve

good filter performance.

5.3.3 Linear Quadratic Gaussian Controller (LQG)

LQR and Kalman filter can be combined together to form the more practical

controller referred to LQG compensator shown in Figure (5.1). In LQG, the controller

output is based on the estimated states instead of the actual states:

U = - K X (5.16)

The LQG controller is based on the linear systems separation principle that allows

separate design of a feedback control gain and of an observer gain to form a dynamic

compensator. Since the Kalman filter is a dynamic system, the LQG is then a linear

dynamic regulator of the same kind used in classical control theory. However, unlike

classical control, the design of LQG is systematic and does not require any knowledge of

the controller structure. This makes LQG very suitable for designing stable control

systems for high order MIMO systems such as the application at hand, nonlinear panel

flutter suppression.

One popular method to design robust LQG compensator is to use the loop-transfer

recovery technique (LQG/LTR). This technique is based on recovering the guaranteed

robustness properties of the LQR (infinite gain margin and 60 degrees phase margin) to

the corresponding LQG controller. This can be achieved by tuning the LQG Kalman filter

weighting matrices Q e and R e to asymptotically recover LQR loop transfer function. One

way to select the Kalman filter weighting is:

R e = m > Q e = v 2 B B T (5.17)

As v goes to infinity, the loop transfer function of LQG controller approaches the loop

transfer function of LQR. However, the value of v should not be increased indefinitely as

this leads to very high Kalman gains. The LQG/LTR is a frequency domain loop shaping

approach that could be used to satisfy specific robustness bounds on the open loop

transfer function.


104

5.3.4 Extended Kalman Filter (EKF)

One major drawback with the application of linear Kalman filter to the problem of

nonlinear panel flutter suppression is that it assumes linearized system equations about

the reference no deflection point. Hence, the nonlinear effects are not considered at all in

the estimation process. As it will be shown in Chapter 6 , this could deteriorate the state

estimation performance for large limit cycle amplitudes. One solution to such a problem

is to use Kalman filter that performs the linearization about a trajectory that is

continuously updated with the estimated states. Such filter is referred to as extended

Kalman filter (EKF) [109], The equations of an EKF for the nonlinear state space model

o f panel flutter, equation (5.3), are given by:

X = A X + BU + K e (Y —CX)

Pe = A P e + Pe A T — PeC T R ~ l CPe + Q e6 e e e v e ( 5 [ g )

K e = P eC T R ; 7

A = A + AAq (X)

where the term AAq (X) represents the nonlinear state space matrix evaluated using the

current state estimate. Unlike the standard Kalman filter, the EKF gain sequence cannot

be predetermined and has to be evaluated on-line. This adds more computational cost but

should not be a problem especially with the advanced and fast real time processors

available today. The EKF deviates from the standard linear Kalman filter because of the

nonlinear feedback of the measurement into the system model and because the various

random variables are no longer Gaussian after undergoing nonlinear transformation.

Thus, the EKF is simply an ad hoc state estimator for nonlinear systems. However, it has

proven to be successful in various applications [109-112],

5.3.5 Nonlinear Controller using EKF and LQR

The EKF can be combined with LQR to form a nonlinear dynamic output

compensator for the control o f nonlinear panel flutter, see Figure (5.2). Such controller is

nonlinear because the estimated state and consequently the control effort, U, is a dynamic

nonlinear function of the measured sensor output. The advantage of this controller over

the LQG controller is the more accurate state estimation for the nonlinear panel flutter

dynamics.


105

5.3.6 Optimal Output Feedback Control

A simpler control strategy for MIMO systems that avoid the need for state

estimation is to design an output feedback compensator of the form:

The output feedback gain matrix, Ky, can be determined using optimal control by

minimizing the quadratic performance index given in equation (5.8). However, this leads

to a feedback control gain that is dependent on the states initial conditions which is

undesirable. This problem can be avoided by minimizing the expected value of the

performance index, E{J}\ hence, the minimization is performed over a distribution of

possible values for the initial conditions. The design equations for optimal output

feedback control are given by [108]:

The expected initial state is usually assumed uniformly distributed on the unit sphere in

order to drive any initial state to zero. It can be seen from equation (5.20) that the

solution for the optimal output feedback gain requires a solution of three coupled

nonlinear algebraic matrix equations. These equations can be solved using the gradient-

based iterative method described in Appendix C [108].

The method adopted for actuator placement in this study is the method of norm of

optimal feedback control gain matrix (NFCG) used for panel flutter suppression by [74]

and [100]. This method determines the effectiveness o f a piezoelectric actuator by using

the norm o f feedback gain designed using LQR and thus is a closed loop criterion.

The higher the value of N F C G , the more effective is the actuator for panel flutter

suppression.

U = —K y Y (5.19)

a J p + PAc + C T K Ty R K y C + Q = 0

A c s + s a J + X o = 0K y = R ' 1 B T P S CT (CSCT )~l

(5.20)

where

A c = A - B K C , (5.21)

5.4 Optimal Placement of Piezoelectric Sensors and Actuators

NFCG = |^ ijQR | (5.22)


106

The NFCG method is extended for optimal sensor location by using the norm of Kalman

filter estimator gain (NKFEG).

N K F E G = \\K Kaiman || (5 .23)

The higher the value of NKFEG, the more effective is the sensor for the problem of state

estimation.

For the problem of panel flutter suppression, a number o f piezoelectric patches

equal to the number of finite element mesh size are used. The effectiveness of each

element as an actuator and as a sensor is determined using the above two criteria. The

shape and location of optimal actuator and sensor is then determined by combining the

most effective elements. Although each piezoelectric patch might act as both actuator and

sensor (self-sensing actuators), the optimal location for actuator and sensor might not be

the same.


107

LQG

U = - K XLQR

System

Kalman Filter

Figure 5.1 Block diagram representation of the LQG controller


108

Nonlinear Dynamic Controller

U = - K X

LQR

NonlinearSystem

Nonlinear State Estimator

(EKF)

Figure 5.2 Nonlinear dynamic compensator for nonlinear

panel flutter suppression using EKF and LQR


109

CHAPTER VI

NONLINEAR PANEL FLUTTER SUPPRESSION RESULTS

6.1 Introduction

Simulation studies for the suppression of nonlinear panel flutter using

piezoelectric material under yawed supersonic flow are presented in this chapter. The

time domain nonlinear finite element modal equations of motion are used to simulate the

panel nonlinear flutter using the Runge-Kutta integration method. The studies presented

have two main objectives. The first objective is the comparison of the controllers

presented in Chapter 5 to determine the effect of different control strategies on the panel

flutter suppression performance. The closed loop system performance is also studied

under system parameter variations. This detailed controller performance study is

performed for the case of zero flow yaw angle. The second objective is controlling

nonlinear panel flutter under yawed supersonic for a specific range of yaw angles (the

range selected in this study is from 0° to 90°).

Panel flutter suppression under yawed supersonic flow is considered for isotropic

square, composite rectangular graphite/epoxy, and isotropic triangular panels. The

properties for both isotropic and composite materials were given in Table 4.1. The main

piezoelectric ceramic used in this study is the isotropic PZT5A. The mechanical and

electrical material properties of PZT5A are given in Table 6.1. The piezoelectric layers

are embedded on the top and bottom layers o f the panel to avoid disturbing the flow filed

over the panel. Each piezoelectric layer will be used as either self-sensing actuator or as

actuator only.

6.2 Square Isotropic Panel

A square simply supported isotropic panel (12x12x0.05 in.) is first used to study

different control methods and to study the control of panel flutter under a range of flow

yaw angles. Piezoelectric layers are embedded on the top and bottom surfaces of the

panel. The piezoelectric thickness is selected as 0.01 in. compared to the panel total

thickness of 0.05 in. Using this thickness, the actuation voltage is then limited to 152


110

Volts (see Table 6.1). The aerodynamic damping coefficient, c a , is assumed 0.01 and a

structural modal damping ratio of 1% is added for all modes.

The panel is modeled using 10x10 finite element mesh and the nonlinear modal

equations of motion are derived using a selected number of modes based on modal

participation and modal convergence. For the case of zero flow yaw angle, four modes in

the flow direction, mode (1,1) to (4,1), are used for modal reduction. This allows for a

low order open loop system and, also, provides reasonably good accuracy for the flutter

limit-cycle oscillation (LCO) amplitude (see modal participation values at a = 0° in Table

4.6). This low order model is used for the detailed controller performance study. For the

case of arbitrary yaw angle, the panel nonlinear modal equations of motion are derived

using 16-mode model, mode (1,1) to (4,4). This makes the state space system of order 32.

6.2.1 Optimal Placement of Self-Sensing Piezoelectric Actuators

The first step is determining the optimal location of the embedded piezoelectric

layers to achieve optimal actuation and optimal sensing. The panel is divided into 10x10

mesh size and is completely covered on top and bottom with piezoelectric layers (total of

100 piezoelectric actuators/sensors). The modal equations of motion are derived using the

16-mode model in order to be able to determine the optimal actuator and sensor locations

for different yaw angles. The linearized system equations of motion about the zero

displacement point, equation (5.5), are used to design both optimal feedback control

(LQR) and optimal observer (Kalman filter). The resulting optimal feedback gain, K , and

observer gain, K e, are matrices with sizes 100x32 and 32x10, respectively. They

represent the feedback and the estimator gain vectors for each patch of the 100

piezoelectric patches used. The methods of NFCG and NKFEG are then used to

determine the effectiveness of each piezoelectric patch for optimal actuation and sensing.

Figures 6.1 and 6.2 show contours of the values of NFCG and NKFEG for the

cases of 0° and 45° flow yaw angles, respectively (A is set to 1000 for this case). These

figures show clearly how the optimal actuator and sensor locations change completely for

different flow angles. This indicates the importance of considering the flow yaw angle

effect for the problem of panel flutter suppression. It is also seen that the optimal sensor

location is at the trailing edge (TE), where the maximum panel deflection point is located,

while the optimal actuator location is at the panel leading edge (LE) in the flow direction.


I l l

The optimal sensor location agrees with the standard structural vibration applications by

placing the sensors at the maximum deflection point which gives minimum noise to

signal ration. However, the optimal actuator location doesn’t agree with the placement

guidelines that are well known for structural vibration, which usually places piezoelectric

actuators at the maximum curvature or maximum strain location. This is due to the

aerodynamic loading involved in the panel flutter problem, which is proportional to the

transverse deflection slopes (vvr and wv). Thus for A > A cr, considering the flutter

deflection shape shown in Figure 4.10, the panel is subjected to a distributed pressure

with maximum amplitude at the panel TE. It is found that the most effective piezoelectric

actuators to counteract this forcing aerodynamic pressure are those located at the panel

LE. This conclusion is in agreement with the experimental testing results performed by

[106],

Figures 6.3 and 6.4 show the selected piezoelectric actuator and sensor location

based on the NFCG and NKFEG methods for 0° and 45° yaw angles, respectively.

Rectangular shaped piezoelectric patches are used to obtain practical sensor and actuator

shapes. However, the optimal sensor and actuator shape can be refined by using more

elements or triangular patches. The optimal sizing of piezoelectric actuators is outside the

scope of this study and hence the size of both pieces is assumed by using the most

effective 12 elements. This assumption is based on the studies performed by [74] which

showed that the optimal size of actuator is between 10% and 20% of the panel area for

nonlinear panel flutter suppression. Since self-sensing piezoelectric actuators are utilized

in this study, the size of both patches located at the LE (optimal actuator) and at the TE

(optimal sensor) is assumed to be the same, as both of them will be used as actuators. In

practice, other types of sensors may be used such as accelerometers, displacement

sensors, or strain gages. A case with single LE actuator and displacement sensors is

considered when designing optimal output feedback control law.

6.2.2 Controller Study

A study of the performance of different controllers described in Chapter 5 is

performed in this section. Two main categories o f controllers are used. The first category

is state feedback using LQR based on estimated states. Linear state estimation using

standard Kalman filter is considered which leads to the LQG controller. In addition.


1 1 2

nonlinear state estimation using extended Kalman filter (EKF) is introduced. The second

category considered is the optimal output feedback control.

The performance of different controllers is compared using their ability to

suppress the panel nonlinear flutter. The measure used to asses performance is the

maximum flutter-free dynamic pressure, A m a x, defined as the maximum dynamic pressure

at which the control system can suppress the LCO of nonlinear panel flutter given the

limitation of the actuator maximum voltage (saturation). This quantity is governed by

controller performance, piezoelectric saturation voltage, and LCO amplitude. To obtain

conservative values for A max, the controller will be activated after the convergence of

LCO.

For the comparison performed in this section, the panel at 0° yaw angle with the

embedded PZT5A piezoelectric self-sensing actuators shown in Figure 6.3 is considered.

The panel is modeled using the 4-mode model (state space model is of 8 lh order). Figure

6.5 shows the change of panel frequencies as the nondimensional dynamic pressure, A ,

increases for the panel with no piezoelectric material and with piezoelectric material

added. It is seen how the addition of piezoelectric material decreases the panel stiffness

compared to the original panel. This occurs because the piezoelectric material has heavier

mass and slightly softer Young’s modulus. Figure 6.6 shows the open loop poles at 3

different values of A . It is seen how increasing A beyond A cr results in 2 unstable poles

(after mode (1,1) and (2,1) coalescence) and increasing it further results in 4 unstable

modes as the other two modes, (3,1) and (4,1), coalesce.

6.2.2.1 LQR Controller

First a full state feedback using LQR is designed for the linearized model

assuming all the states available. As discussed in Chapter 5, this is not a practical control

method unless it is combined with state estimation. The state-weighting matrix, Q , is

selected using the energy method and the control-weighting matrix, R , is selected as a

constant multiplied by the identity matrix.

K 0 .0 M

Different values of ra re tested until satisfactory performance is achieved using r = 1000.

Figure 6.7 shows the performance of the LQR control for nonlinear panel flutter

Q = R = r [ l ] (6 . 1)


113

suppression at A = 1500. The controller is activated at t = 100 msec and it suppresses the

LCO using the two self-sensing actuators at LE and TE within few cycles. A „ iax using this

controller was found to be 1760 compared to 512 for the original panel and to 449 for the

panel with embedded piezoelectric material and with no control applied.

6.2.2.2 LQG Controller

A Kalman filter estimator is designed based on the system linearized equations of

motion. Many values are used for the design parameters Q e and R e to arrive at

satisfactory estimator performance. The performance of the Kalman filter estimation is

shown in Figures 6.8 and 6.9 for A = 1000 and 2000 respectively. It is seen how the

estimated LCO amplitude degrade as A increases. This is expected as the Kalman filter

uses a linear model that does not consider the nonlinear effects at all, so it is trying to

estimate a nonlinear process using a linear model. As A increases, the effect of

nonlinearity increases and hence the estimated LCO amplitude using Kalman filter is not

accurate. The design parameters of the Kalman filters are selected as: Q e = [/], R e =

[C][C]T.

Figure 6.10 shows that, unlike the LQR, the LQG controller cannot suppress the

flutter LCO at A = 1500. Comparing this figure to the LQR performance, shown in Figure

6.7, shows how the introduction of linear state estimation degrades the control system

performance. The achieved A m a x using LQG is 920 compared to 1760 using LQR control,

which is about 50% performance reduction due to linear state estimation. Figure 6.11

shows the performance of the LQG controller at the maximum flutter-free dynamic

pressure.

6.2.2.3 Nonlinear Output Controller (EKF+LQR)

An extended Kalman filter estimator (EKF), which takes into account the system

nonlinear dynamic by linearizing about the estimated state vector, is implemented using

the same parameters as those of a Kalman filter (Q e, R e ). The performance of state

estimation using EKF is shown in Figure 6.12 at A — 2000. Comparing this to Figure 6.9

shows how superior the performance of EKF is compared to the standard Kalman filter. It

has to be noted that, in practice, the estimation performance will be further degraded due

to sensor noise and system model uncertainty.


114

Combing the EKF with LQR control results in a dynamic nonlinear compensator.

Figure 6.13 shows the performance of the EKF+LQR for nonlinear panel flutter

suppression at A = 1500. Using this controller, A m ax was found to be 1750. To show the

performance of EKF+LQR control even with uncertainty in the model nonlinear matrix

used by the EKF, the controller is used with +/-25% uncertainty in the system nonlinear

matrix, A A q , used in propagating the state estimate, see equation (5.16). Figure 6.14

shows that the flutter LCO can be successfully suppressed using the EKF+LQR

controller with +/-25% uncertainty in A A q . Therefore, using nonlinear state estimation

instead of the standard linear state estimation for the problem of nonlinear panel flutter

gives much better results even with some uncertainty in the nonlinear model matrix.

In all the controllers above, LQR, LQG, and EKF+LQR, the simulation results

indicated that the self-sensing piezoelectric patch at the TE is not as effective as the LE

patch from actuation point of view. However, the TE piezoelectric patch is acting as the

optimal sensor to achieve better state estimation. Without this sensor, i.e., if the LE

sensor is only used, the estimation performance will be degraded dramatically due to the

high noise to signal ratio for the LE sensor resulting from smaller displacement at this

location.

6.2.2.4 Optimal Output Feedback Controller

The iterative algorithm given in Appendix C is used to solve the coupled

nonlinear equations given by equation (5.20) for optimal output feedback gain matrix. An

initial stabilizing gain, K a , has to be first determined using some trail and error. The

weighting matrices Q and R are the same as for the LQR case. For the panel

configuration of 2 self-sensing piezoelectric actuators, the resulting output feedback gain

matrix is of size 2x2. Using this configuration, an output feedback gain was successfully

designed to suppress flutter LCO up to A m ax = 1000. Figure 6.15 shows the performance

o f the optimal output feedback controller at the maximum flutter-free dynamic pressure.

It is noticed that this controller introduces less damping to the system compared to the

full state feedback control. In practice, low pass filters have to be used to filter out the

feedback signals from the sensors. This could degrade the optimal feedback performance

even more.


115

Another panel configuration that uses only the LE actuator and uses two

displacement sensors located at (a/2, a/2) and (3a/4, a/2) is considered. The output

feedback gain for this configuration is o f size 1x2. Figure 6.16 shows that the optimal

output feedback can suppress LCO up to A mclx = 1100 for this case.

In conclusion, the optimal output feedback performance for panel flutter

suppression is less than using EKF+LQR control. However, the optimal output feedback

is much simpler for practical implementation as it does not require the on-line adaptation

needed for EKF. The achieved values of A m ax for the different controllers used are

summarized in Table 6.2.

6.2.2.5 Controller Robustness

A study for the effect of system parameter variation on the performance of both

EKF+LQR and optimal output feedback controllers is given in this section. First with

regard to the dynamic pressure A , it was found that for LQR and optimal output feedback,

the feedback gain determined at A nw x suppresses LCO for all values of A < A m a x.

However, for accurate state estimation X needs to be known since the system model is a

function of A . Figure 6.17 shows the effect of +25% mismatch in A between design model

and reality (simulation model) for both LQR and EKF+LQR controllers when the

simulation model has A = 1200. It is seen how the performance of EKF+LQR is degraded

in this case, mainly due to inaccurate state estimation.

The second type of system parameter variation considered is in the panel stiffness.

In section 6 .2.2.3, it was shown that, the EKF+LQR controller has a satisfactory

performance with +/-25% uncertainty in the design model nonlinear stiffness matrix. In

this section a +/-25% variation in the modal linear stiffness is considered, which

corresponds to about +/-13% uncertainty in the panel natural frequencies. Figure 6.18

shows the effect of +/-25% variation in modal linear stiffness using EKF+LQR controller

at A — 1500. It is seen that the LCO cannot be suppressed with such system parameter

variation. The performance of EKF+LQR is degraded with parameter variations because

both the state estimation and the state feedback design are affected by the linear stiffness

variation. Figure 6.19 shows the same effect for the optimal output feedback controller at

A = 1100 (using the configuration of single LE actuator and 2 displacement sensors). The


116

optimal output feedback controller is shown to be more robust to linear stiffness variation

than the EKF+LQR controller.

6.2.3 Panel Flutter Control with Yawed Flow

In this section, nonlinear panel flutter suppression with flow yaw angle is

considered. The nonlinear dynamic output compensator (EKF+LQR) is used in this study

as it provides the best performance. First, the flow yaw angle, a , is considered as simply

a perturbation that is not considered in the design at all, i.e., not considered in the

piezoelectric placement or in the controller design. Figure 6.20 shows the performance of

the controller designed for a = 0° when the actual flow is at a = 45° and A = 800. The

results clearly indicate that the flutter LCO cannot be suppressed for a = 45° using the

control system designed for a = 0° even at such low dynamic pressure. This shows that

flow yaw angle has to be considered in the design of active control systems for nonlinear

panel flutter suppression. Figure 6.21 shows the effect of flow angle on the optimal

location of self-sensing piezoelectric actuators for a = 0, 15, 30, 45, and 90° flow yaw

angles. The flow LE piezoelectric patch is the optimal actuator location d6etermined

using NFCG while the TE patch is the optimal sensor location determined using NKFEG.

It is seen how the change of flow angle has a major effect o f the optimal actuator and

sensor locations.

The objective is to design active control system including piezoelectric placement

for the panel knowing that the panel will be subjected to a specific range of flow yaw

angles. The range assumed in this study is from 0 to 90°. Performing optimization for

piezoelectric placement over a range of a is not a simple task. The methodology

proposed in this study is to optimize the piezoelectric sensor and actuator placement for

different angles within the specified range, as was shown in Figure 6.21, then group the

resulting optimal location together to find a single piezoelectric configuration that works

over the entire range of angles.

Figure 6.22 shows optimal actuator and sensor location that covers all the yaw

angles from 0 to 90°. The optimal actuator consists o f 2 LE piezoelectric pieces in the

directions normal to 0 and 90° while the optimal sensor consists o f 2 TE piezoelectric

pieces in the directions normal to 0 and 90°. As mentioned before, the piezoelectric

patches required for optimal sensing can be replaced by other types of sensor such as


117

strain gages or can be smaller in size. However, since they will be used as actuators too,

piezoelectric patches o f the same size are used. Thus, the configuration o f Figure 6.22

results in an outer square covering 48% of the panel surface with embedded PZT5A. The

embedded piezoelectric is divided into 4 equal self-sensing actuators referred to as: OLE

and 90LE for the leading edge patches perpendicular to the 0 and 90° flow directions, and

OTE and 90TE for the trailing edge patches perpendicular to the 0 and 90° flow

directions. Examining this configuration further, it is seen that it does not only cover yaw

angles from 0 to 90° but due to the panel symmetry and the usage o f self-sensing

actuators, this configuration will cover all angles from 0 to 360°.

The performance of the EKF+LQR controller for nonlinear panel flutter

suppression with yawed flow is now considered. Three different configurations for

piezoelectric placement are considered for comparison. The first and second

configurations are simply using the optimal placement determined at a = 0° and 45°,

respectively, while the third configuration uses the optimized piezoelectric over the range

o f 0 to 90° with 4 self-sensing actuators, as was shown in Figure 6.22. Figure 6.23 shows

a comparison of the achieved A m ax using these 3 configurations for flow yaw angles

changing from 0 to 90°. It is seen that the piezoelectric optimized for 0° flow angle

provides acceptable performance up to a = 45° with A m axl A cr - 3.03 ( A c r = 512), but

A m a J A c r decreases to about 1.56 at a = 90° compared to 3.41 at a = 0°. Therefore, this

configuration is not a good candidate for yaw angles from 0 to 90°. However, it provides

acceptable performance from 0 to 45° and consequently from -4-5° to 45° due to the panel

symmetry. The second configuration, piezoelectric optimized for a = 45°, is shown to

provide good flutter suppression performance over the entire range yaw angles from 0 to

90°. The best performance is achieved at a = 45° with A m axl A cr = 3.71 while A rnax! A cr =

3.0 at a = 0° and 90°. The third configuration with 4 self-sensing piezoelectric actuators

is shown to provide the best performance of all configurations with a minimum of

A m a x /A c r = 3.52 over the entire range of yaw angles. However, the second configuration

uses only 24% covered panel compared to 48% for the third configuration. Figures 6.24

through 6.26 show the time history of maximum LCO and the control inputs to the four

self-sensing piezoelectric actuators using the EKF+LQR controller at A = 1500 for a =


118

0°, 45°, and 90°, respectively. They clearly indicate the effectiveness of proposed

controller and piezoelectric configuration in suppressing nonlinear panel flutter at

different flow yaw angles.

6.3 Composite Panel

Nonlinear panel flutter suppression with yawed supersonic flow is also considered

for a rectangular (15x12x0.048 in.) simply supported graphite/epoxy [0/45/-45/90]s

panel. PZT5A actuators that have the thickness of 0.006 in. are embedded at the top and

bottom layers, i.e., replacing the 0° layer. The maximum actuator applied voltage is

limited to 91.2 Volts in this case. The panel is modeled using 10x10 MIN3 finite element

mesh and the nonlinear modal equations of motion are derived using the lowest 16

modes. The aerodynamic damping coefficient, c a, is assumed 0.01 and the modal

structural damping ratio = 1%.

Adding too much piezoelectric material for this case will largely change the panel

characteristics as it replaces the 0° layer and hence change the original panel directional

properties. In addition, PZT5A mass density is very high compared to graphite/epoxy

(about 5 times higher). To avoid adding too much piezoelectric material and changing the

panel characteristics, piezoelectric material is only used as actuator for this panel and

displacement sensors are used to provide output measurements for the estimation process.

Similar to the isotropic panel case, the optimal actuator location is determined by

dividing the panel into 10x 10 piezoelectric actuator elements at the top and bottom then

using LQR state feedback gain matrix from these actuators to select the elements with the

highest NFCG. Figure 6.27 shows optimal piezoelectric actuator using this method for a

— 0, 15, 30, 45, and 90° flow yaw angles at A. = 800. Considering a yaw angle range from

0 to 90°, the approximate shape of optimal piezoelectric actuator that cover all angles

within this range is shown in Figure 6.28. It is basically the union of all elements at

different angles shown in Figure 6.27. The piezoelectric actuator shown in Figure 6.27 is

divided into two separate actuators one perpendicular to the 0° flow direction and the

other to the 90° flow direction. Three displacement sensors are used at the locations of

(3a/4, b l 2), (3a/4, 3 b / 4 ) , and { a l l , 3b/4) corresponding to the approximate location of

maximum deflection for 0°, 45°, and 90° yaw angles, respectively.


119

Like to the isotropic panel, the performance of the EKF+LQR controller for

nonlinear panel flutter suppression is studied using three different configurations for

piezoelectric actuator placement: the optimal actuator for the case o f a = 0°, the optimal

actuator for a = 45°, and the optimized actuator over the range of 0 to 90° with 2

independent actuators, as shown in Figure 6.28. Figure 6.29 shows a comparison of the

achieved A m ax using these 3 configurations for flow yaw angles changing from 0 to 90°.

The critical dynamic pressure changes largely with flow angle for the original panel with

no piezoelectric actuators added as was shown in Figure 4.12 ( A cr= 315, 250, and 185 for

flow angles o f 0°, 45°, and 90°, respectively). The conclusion drawn from this figure are

very similar to those of the isotropic panel case and are summarized as follows:

• The actuator optimized for 0° flow angle does not provide good flutter

suppression performance over the range of 0 to 90°. However, it is reasonably

good for yaw angles from -45 to 45°.

• The best performance over the flow angles range of [0,90°] is achieved using the

actuator optimized for this range with 2 LE actuators (OLE and 90LE).

• The actuator optimized for 45° angle provides reasonably good performance for

all angles from 0 to 90° and has the advantage of being half the size of the

actuator optimized for all angles from 0 to 90°.

Thus, the panel maximum flutter free dynamic pressure can be increased to a minimum

of 750 over all yaw angles from 0 to 90° using the actuator optimized for 45° or to a

minimum of 900 using the actuator optimized for all angles from 0 to 90°. This is

compared to the minimum flutter-free dynamic pressure of 185 occurring at a = 90° for

the original panel. Figures 6.30 through 6.32 show the time history of maximum LCO

and the control inputs to the two piezoelectric actuators using the EKF+LQR controller at

A = 900 and for a = 0°, 45°, and 90° respectively. It shows the effectiveness of proposed

controller and piezoelectric configuration in suppressing nonlinear panel flutter at

different flow yaw angles for composite panels.

6.4 Triangular Panel

The triangular (12x12x0.05 in.) clamped isotropic panel studied in Chapter 4 is

considered for nonlinear panel flutter suppression. PZT5A layers with a thickness of 0.01


120

in. are used. The panel is modeled using 10x10 MIN3 finite element mesh and the

nonlinear modal equations of motion are derived using the lowest 16 modes. The flow

yaw angles range selected for this panel is between 90° and 180°. This range is selected

since it encompasses the worst case yaw angle where the flow is parallel to the tilted edge

(or= 135° or 315°), as was shown in Chapter 4.

The optimal actuator location is determined by dividing the panel into 100

triangular piezoelectric elements at the top and bottom then using the NFCG method.

Figure 6.33 shows optimal piezoelectric actuator location using this method for different

yaw angles between 90° and 180°. An approximate shape and location of a single

piezoelectric actuator for the flow range from 90° to 180° is determined by combining the

optimum actuators at different angles within this range of yaw angles as given in Figure

6.34. A single displacement sensor is used at (0.2a, 0.46). This sensor location is

determined based on the locations o f maximum panel deflection at different flow angles

which are (0.2a, 0.36), (0.2a, 0.46), and (0.2a, 0.56) for a = 90, 135, and 180°,

respectively.

The performance of the EKF+LQR controller for nonlinear panel flutter

suppression is given in Figure 6.35. The panel maximum flutter free dynamic pressure is

increased to a minimum of 6350 compared to a minimum of 2620 for the uncontrolled

panel. The piezoelectric material and active control system for this panel is not as

effective as the case of square isotropic panel (3.5 times increase in X cr for the square

isotropic panel versus only 2.4 increase for the triangular panel). This is mainly due to the

smaller size and clamped boundary conditions of the triangular panel, which make it

stiffen

6.5 Summary


piezoelectric materials and different optimal control strategies are presented in this

chapter. The control strategies considered include LQG controller, LQR combined with

extended Kalman filter (EKF), and optimal output feedback. The nonlinear dynamic

output compensator compromised of LQR and EKF results in a 3.5 times increase in the

critical dynamic pressure compared to only 1.8 times for the LQG controller. Using

optimal output feedback controller, the panel critical dynamic pressure was increased


121

about 2 times. However, the optimal output feedback is much simpler for practical

implementation as it does not require the on-line adaptation needed for EKF. Suppression

of nonlinear panel flutter under yawed supersonic flow is considered using the LQR and

EKF nonlinear controller. The NFCG and NKFEG methods are used to determine the

optimal location of piezoelectric actuators and sensors, respectively. Optimal actuator and

sensor location for a range of flow yaw angles is determined by grouping the optimal

locations for different angles within the specified range. Using this method with four self

sensing actuators, the critical flutter boundary was increased 3.5 times over the entire

range of yaw angles from 0 to 360° for square isotropic panel. For a rectangular

graphite/epoxy composite panel, the critical flutter dynamic pressure is increased to a

minimum of 900 over all yaw angles from 0 to 90° compared to 185 for the original

uncontrolled panel. Results for a clamped triangular isotropic panel showed that the

critical flutter dynamic pressure can be increased about 2.5 times for a flow yaw angle

range from 90 to 180°.


1 2 2

Table 6.1 Mechanical and electrical properties PZT5A piezoelectric ceramics

PZT5A

H, 9.9 Msi (69 Gpa)

e 2 9.9 Msi (69 Gpa)

G [2 3.82 Msi (26.3 Gpa)

g 23 3.82 Msi (26.3 Gpa)

d3i -6.73x10 'y in/V (-171X101- m/V)

d32 -6.73x10 v in/V (-171X101" m/V)

P 0.72x1 O'J lb-sec‘/in.4(7700 K g/m ’)

Vi2 0.31*

he Same as layer thickness

Emax 15240 V/in (600 V/mm)

* hc: Electrode spacing

Table 6.2 Comparison of different controllers performance

for nonlinear panel flutter suppression

Controller ■max

LQR 1760

LQG 920

EKF+LQR 1750

Optimal OFB 1000-1100


123

0.8

0.6

0.4

0.2

0.2 0.60.4 0.8x/a

(a)

0.8

0.6

£0.4

0.2

0.2 0.60.4 0.8x/a

(b)

Figure 6.1 Contours of (a) NFCG and (b) NKFEG for simply

supported square isotropic panel at 0° flow angle


124

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8x/a

(a)

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8x/a

(b)

Figure 6.2 Contours of (a) NFCG and (b) NKFEG for simply

supported square isotropic panel at 45° flow angle


125

Optimal Actuator Optimal Sensor

Figure 6.3 Selected self-sensing piezoelectric actuators placement and size for optimal

actuation and optimal sensing on square isotropic panel at 0° flow angle


126

Optimal Actuator Optimal Sensor

Figure 6.4 Self-sensing piezoelectric actuators placement and size for optimal actuation

and optimal sensing on square isotropic panel at 45° flow angle


Freq

uenc

y (H

z)

127

600

500

400

300

2 0 0

100No Piezo With Piezo

0 500 15001000 2000D yn am ic P re ssu re

Figure 6.5 Variation of First 4 linear modes versus X for isotropic

panel with and without added piezoelectric material


Imag

inar

y

128

X X = 0, + X = 1000, V X = 2000

40 0 0

3 0 0 0

2000

1000

1000

-he-2000

-3 0 0 0

-4 0 0 01 0 0 ' 1-----------------------1000 -500 0 500 1000

Real

Figure 6.6 Open loop poles for isotropic square panels with embedded piezoelectric

material at different values of nondimensional dynamic pressure


2

1

0

1

2

80 90 100 110 120

a 100ci.GLE

't e

© uu

1-100

9080 100 110 120Time (msec)

Figure 6.7 Limit-cycle amplitude and control inputs time history

for square isotropic panel using LQR control at X = 1500


Permission of the

1.5

0.5

£ -0.5

-1.5

-2.5

85n " « (m sec)

100

F i g u r e 6 . 8 CnmmPanson be,ween

c o a m p l i t u d e us/no amplitude and tan fi,te r« A = 1000

“ pyright owner F„Bh

131

I++

ActualEstimated

75 8070 85 90 95 100Time (msec)

Figure 6.9 Comparison between actual LCO amplitude and estimated

LCO amplitude using Kalman filter at X = 2000


Con

trol

Inp

ut

132

-2 •

80 90 100 110 120 130 140 150

100

LE't e-100 ■

80 90 100 110 120 130 140 150Time (msec)

Figure 6 .10 Performance of LQG controller at X = 1500


Con

trol

Inp

ut

133

2

I

0

1

280 100 120 140 160 180 2 0 0

100

£-100

80 100 120 140 160 180 200Time (m sec)

Figure 6.11 Performance of LQG controller at the maximum

suppressible dynamic pressure, Xmax = 920


134

2.5ActualEstimated

1.5

0.5-c■ac

£0.5

-1.5

-2.590 92 94 96 98 100

Time (m sec)

Figure 6.12 Comparison between actual and estimated LCO amplitude

using extended Kalman filter at k = 2000


Con

trol

Inp

ut

135

2

1

0

1

2

80 90 100 110 120 130 140 150

£100

-100

80 90 100 110 120 130 140 150Time (msec)

Figure 6.13 Performance of LQR+EKF nonlinear output compensator at X = 1500


136

21012

80 90 120100 1 1 0 130 140 150

LETE

1 00

0

- 1 0 0

80 90 120100 1301 1 0 140 150T i m e ( m s e c )

(a)

21012

80 90 100 1201 10 130 140 150

LETE

100a.s

- 1 0 0

80 90 120100 1 1 0 130 140 1 50T i m e ( m s e c )

(b)

Figure 6.14 Performance of LQR+EKF nonlinear output controller at X = 1500

with (a) -25% and (b) +25% uncertainty in the model nonlinear stiffness matrix


Con

trol

Inp

ut

137

2

1

0

1

280 100 120 140 160 180 2 0 0

100

LE'TE

100

80 100 120 140 160 180 2 0 0Time (msec)

Figure 6.15 Performance of optimal output feedback controller

using two self-sensing actuators at Amax = 1000


Con

trol

In

put

138

2

1

0

-1

280 100 120 180140 160 200

100

80 100 120 140 160 180 200Time (msec)

Figure 6.16 Performance of optimal output feedback controller using single

leading edge actuator and two displacement sensors at A.max = 1100


139

LQRLQR+EKF

csE

£

70 80 90 100 110 120 130 140 150lim e (msec)

Figure 6.17 Effect of +25% mismatch in X between design model and simulation model

on LQR and LQR+EKF control performance at X = 1200


140

3

2

1

0

-1

-2

-3

80 100 120 140 160 180 200Time (msec)

Figure 6.18 Effect of design model linear stiffness variation on

The LQR+EKF controller performance at X = 1500


+25%-25%1.5

0.5

E£

-0.5

-1.5

80 100 120 140 160 180 200l im e (msec)

Figure 6.19 Effect of design model linear stiffness variation on optimal

output feedback controller performance at X = 1100


2

1

0

1

280 90 100 110 120 130 140 150

s 100

0

LE't e-100

80 90 100 120110 130 140 150Time (m sec)

Figure 6.20 Performance of EKF+LQR controller designed for

zero flow angle at X = 800 and 45 deg flow angle


a = 0° a = 1 5 °

a = 45°

S i l l

a = 90°

Figure 6.21 Optimal actuator and sensor placement at different flow

angles from 0 to 90° for square isotropic panel


144

Figure 6.22 Placement o f four self-sensing piezoelectric actuators for optimal actuation

and optimal sensing over the range of [0, 90°] flow angle for square isotropic panel


145

2200

A.,max

1600

No PiezoPiezo optim ized for 0 deg Piezo optim ized for 45 deg Piezo optimized for [0-90] deg range

30 40 50 60Flow Yaw Angle (deg)

Figure 6.23 Comparison of panel flutter suppression performance using LQR+EKF

control at different flow yaw angles and using different piezoelectric placement

configurations for a square isotropic panel


Con

trol

Inp

ut

Con

trol

In

put

146

2

0

280 90 100 110 120 130 140 150

100OLE'oTE

-100

80 90 100 110 120 130 140 150

10090LE'90TE

80 90 100 110 120 130 140 150Time (msec)

Figure 6.24 Performance of LQR+EfCF controller for square isotropic panel with 4 self

sensing piezoelectric actuators at 0° flow yaw angle and X = 1500


Con

trol

In

put

Con

trol

Inp

ut

147

2

0

280 90 100 110 120 130 150140

100OLE'oTE

-100

80 90 100 110 120 130 140 150

10090LE'90TE

-100

80 90 100 110 120 130 140 150Time (msec)

Figure 6.25 Performance of LQR+EKF controller for square isotropic panel with 4 self

sensing piezoelectric actuators at 45° flow yaw angle and A. = 1500


Con

trol

In

put

Con

trol

Inp

ut

148

2

0

280 90 100 110 120 130 140 150

100OLE'oTE

80 90 100 110 120 130 140 150

10090LE'90TE

9080 100 110 120 130 140 150Time (msec)

Figure 6.26 Performance of LQR+EKF controller for square isotropic panel with 4 self

sensing piezoelectric actuators at 90° flow yaw angle and A. = 1500


149

a = 0 ° a = 1 5 °

a = 30° a = 45°

a = 90°

Figure 6.27 Optimal actuator placement at different flow angles

for [0/45/-45/90]s composite rectangular panel


150

Figure 6.28 Optimal placement of 2 embedded piezoelectric actuators that cover

flow angles from 0° to 90° for [0/45/-45/90Js composite rectangular panel


151

1600

1400

1200

100

-B-

'-max

0

No PiezoPiezo optim ized fo r 0 degPiezo optim ized fo r 45 degPiezo optim ized fo r [0,90] deg range

0 10 20 30 40 50 60F low Yaw Angle (deg)

70 80 90

Figure 6.29 Comparison of panel flutter suppression performance using LQR+EKF

control at different flow yaw angles and using different piezoelectric actuator

configurations for [0/45/-45/90]s rectangular composite panel


Con

trol

Inp

ut

152

2

0

2

80 90 100 110 120 130 140 150

100

90

-50

-10080 90 100 110 120 130 140 150

Tim e (m sec)

Figure 6.30 Performance of LQR+EKF controller for rectangular

composite panel at 0° flow yaw angle and X = 900


Con

trol

In

put

153

2

0

2

80 90 100 110 120 130 140 150

100

50 90

-50

-10080 90 100 110 120 130 140 150

Tim e (m sec)


composite panel at 45° flow yaw angle and A = 900


1

0.5

0

-0.5

180 90 100 110 130120 140 150

100

g, 50 90

.© -50

-10080 10090 110 120 140 150130

Tim e (m sec)


composite panel at 90° flow yaw angle and k = 900


155

t v\ \ V\ \ [x \\ \ \ \\ \ V \ \\ \ \ \\ \ \ \

a = 90°

a = 120°

a = 150°

a = 105°

a = 165°

a = 180°

Figure 6.33 Optimal actuator placement at different flow angles

for clamped triangular isotropic panel


156

Figure 6.34 Optimal placement of a single self-sensing piezoelectric actuator that

approximately cover all angles from 90° to 180° for triangular clamped isotropic panel


157

7000

6000

5000max

■H— No Piezo-i— Piezo optim ized for 90 deg •©— Piezo optim ized for 135 deg

Piezo optim ized for [90- I80]deg range

4000

300

200090

Flow Yaw Angle (deg)

Figure 6.35 Performance of the LQR+EKF controller in suppressing nonlinear panel

flutter using different piezoelectric actuator configurations

for the clamped triangular isotropic panel


158

CHAPTER VII

SUMMARY AND CONCLUSIONS

A coupled structural-electrical modal finite element formulation for composite

panels, with integrated piezoelectric sensors and actuators, is presented and used to

analyze nonlinear supersonic panel flutter considering the effect of airflow yaw angle and

the effect o f additional high acoustic pressure loading. The finite element formulation is

also used for nonlinear panel flutter suppression with yawed airflow using optimal

control methods. The first-order shear deformation theory is used for laminated

composite panels and the von-Karman nonlinear strain-displacement relations are

employed for large deflection response. Structural-electrical coupling is considered using

the linear piezoelectricity constitutive relations. The first-order piston theory and

simulated Gaussian white noise are employed to model supersonic aerodynamic and

acoustic pressures, respectively. The coupled nonlinear equations of motion are derived

using the three-node triangular MIN3 plate element with improved transverse shear.

Additional electrical DOF per each piezoelectric layer is used to handle piezoelectric

sensors and actuators. Thus, the modified MIN3 element is a coupled structural-electrical

shear-deformable element for the nonlinear analysis o f smart composite structures. The

system equations of motion in the structure node DOF are transformed into the modal

coordinates using the panel linear vibration modes to obtain a set of nonlinear dynamic

modal equations o f lesser number. Modal participation is defined and used to determine

the number of modes required for accurate solutions. A new and efficient solution

procedure is presented using the LUM/LTF approximate method to solve the reduced

nonlinear modal equations for nonlinear panel flutter limit-cycle response. The presented

solution procedure has the advantage of using much less computational effort than

solving the system equations of motion in the structure node DOF.

The presented finite element modal formulation is validated by comparison with

other finite element and analytical solutions. Analysis results for the effect of arbitrary

flow yaw angle on nonlinear supersonic panel flutter for isotropic and composite panels

are presented using the modal LUM/NTF solution method. Results showed that the flow


159

yaw angle completely changes the shape of the Iimit-cycle deflection. It also showed that

the effect of the yaw angle is a very important parameter especially for composite panels

where the flow direction may increase or decrease the nondimensional dynamic pressure

at Fixed limit-cycle amplitude depending on the panel lamination. The effect of combined

supersonic aerodynamic and acoustic pressure loading on the nonlinear dynamic response

of isotropic and composite panels is also presented. It is found that for panels at

supersonic flow, only acoustic pressure (sonic fatigue) is to be considered for low

dynamic pressures ( A « A cr ) and both acoustic and aerodynamic pressures must be

considered for significant and high aerodynamic pressures.


piezoelectric self-sensing actuators and using different optimal control strategies are

presented. The control strategies considered include LQG controller, LQR combined with

extended Kalman Filter (EKF) for nonlinear systems, and optimal output feedback. The

LQG controller performance was found to be much worse than the corresponding LQR

controller performance. This is mainly due to the use of linear Kalman filter to estimate

the states of nonlinear flutter dynamics. The state estimation is improved by using EKF.

The nonlinear dynamic output compensator compromised o f LQR control and EKF gives

much better performance for nonlinear panel flutter suppression with about 3.5 times

increase in the critical dynamic pressure compared to 1.8 times using LQG controller. By

using optimal output feedback controller the panel critical dynamic pressure was

increased about 2 times which is less than that of EKF+LQR controller. However, the

optimal output feedback is much simpler for practical implementation as it does not

require the on-line adaptation needed for EKF.

Nonlinear panel flutter suppression with airflow yaw angle is considered using the

EKF+LQR controller. Closed loop criteria based on the norm of feedback control gains

(NFCG) for actuators and on the norm of Kalman Filter estimator gains (NKFEG) for

sensors are used to determine the optimal location of self-sensing piezoelectric actuators

for different yaw angles. Optimal actuator and sensor location for a range of flow yaw

angles is determined by grouping the optimal locations for different angles within the

specified range. Using this method with four self-sensing actuators, the critical flutter

boundary was increased about 3.5 times over the entire range o f yaw angles from 0 to


160

360° for square isotropic panel. For rectangular graphite/epoxy composite panel with two

actuators, the panel critical flutter dynamic pressure is increased to a minimum of 900

over all yaw angles from 0 to 90° compared to 185 for the original uncontrolled panel.

Results for a triangular isotropic panel with clamped boundaries showed that the critical

flutter dynamic pressure can be increased about 2.5 times for a flow yaw angle range

from 90 to 180° using the same methodology with a single piezoelectric actuator.

The main contributions of this research can be summarized as follows:

• Analysis o f nonlinear panel flutter of composite panels with yawed supersonic

flow using finite element method.

• Analysis of nonlinear composite panels response under combined aerodynamic

and high acoustic pressure loading.

• The consideration of state estimation problem for nonlinear panel flutter

suppression and the use of nonlinear state estimation based on EKF for improved

controller performance.

• Nonlinear flutter suppression o f isotropic and composite panels with yawed

supersonic flow.

Other minor contributions include, the derivation of coupled structural-electrical

nonlinear MIN3 element for composite panels, the modal LUM/NTF solution method,

optimal sensor location based on the NKFEG method, and the study of triangular panels.

Future extensions to the current research may include adding thermal load effects,

using more rigorous optimization method for optimal piezoelectric sensors and actuator

location such as genetic algorithms or gradient methods, and using robust and nonlinear

control strategies. In addition, a feasibility study is required to compare the performance

o f nonlinear panel flutter suppression using piezoelectric actuators and using other types

o f smart and adaptive structures such as shape memory alloys and active constrained

layer damping. Finally, experimental validation is highly desirable to verify the

performance o f nonlinear panel flutter suppression using the proposed methods.


161

REFERENCES

1. Dowell, E. H., “Panel flutter: a review of the aeroelastic stability of plates and

shells,” A I A A J o u r n a l , Vol. 8, No. 3, 1970, pp. 385-399.

2. Mei, C., Abdel-Motagaly, K., Chen, R., “Review of nonlinear panel flutter at

supersonic and hypersonic speeds,” A p p l i e d M e c h a n i c s R e v i e w , Vol. 52, No 10,

1999, pp. 305-332.

3. Gray, C. E., Jr and Mei, C. “Large amplitude finite element flutter analysis of

composite panels in hypersonic flow,” A I A A J o u r n a l , Vol. 31, No. 6 , 1993, pp.

1090-1099.

4. Bismarck-Nasr, M. N., “Finite element analysis of aeroelasticity of plates and

shells,” A p p l i e d M e c h a n i c s R e v i e w , Vol. 45, No 12, 1992, pp. 461-482.

5. Zhou, R. C., Xue, D. Y. and Mei, C., “On analysis of nonlinear panel flutter at

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nonlinear estimation,” P r o c e e d i n g s o f t h e 1 9 9 6 I E E E i n t e r n a t i o n a l C o n f e r e n c e

C o n t r o l A p p l i c a t i o n s , Dearborn, MI, Sept. 1996, pp. 338-343

112. Ewing, C. M., “An analysis of the state dependent Riccati equation method

nonlinear estimation technique,” P r o c e e d i n g s o f t h e A I A A G u i d a n c e , N a v i g a t i o n ,

a n d C o n t r o l C o n f e r e n c e a n d E x h i b i t , Denver, CO, August 2000, pp. 1093-1103,

113. Santos, A., and Bitmead, R. R., Nonlinear control for an autonomous underwater

vehicle (AUV) preserving linear design capabilities,” P r o c e e d i n g s o f t h e 3 4 th

I E E E C o n f e r e n c e o n D e c i s i o n a n d C o n t r o l , New Orleans, LA, Dec. 1995, pp.

3817-3822.


172

APPENDIX A

M I N 3 E L E M E N T S T R A I N I N T E R P O L A T I O N M A T R I C E S

Expressions for the strain interpolation matrices as function of element geometry

and element area coordinates are given in this Appendix for the MIN3 triangular element.

414 2

h )

i2A

-t 2-v 3 - -r 3-v 2 V'2 - >’3 X3 — *2

.v3y 1 - .v ly3 v3 - y l - , r 3

_-vl-v2 ~-v2-vI -vl - -v2 x 2 ~ x l

x u = x i ~ x j

y n = y i -> ’/

[ Q J = - L-2A

-V230

Vl30

V120

_x 32 -v13 r 21

0

-v32

V23

0

-v13-v13

0

x 2 lV[2

[ C v b ] - 2 A>’23 >’3 1 ^12

_-r 32 -r 13 -r 21_

0 0 0 y*>3

v32 -r 13 -r 21 0

_.v 23 -v 31 -VI2 -v32

Vl30

-v 12

0.v

[ c 1= —IV iffy/ J „ ,2A-11 C12 C 13 C 14

13 *21 _

C i5 C 16

C 2 I ^22 ^-23 ^ 24 ^ 25 ^26

2A*32

_-v 23

*13V31

* 2 1

V12.

[c 1= —L 2A

C21 + C-22 +<s 2. C l l C 12

^ 2 3 + 4 3 ^24 C 25

Q 3 C l 4 + 4 l C 15+ £ 2

where

C[ 1 - 0 . 5 ( y l2^2 ~ 3'3I^3 )3,23

C 12 = 0 -5 ( y 23^3 ~ >12^1 ).v 31

(A .l)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6 )

(A.7)

C 26

C [6 + lo 3 .

(A.8 )


c 13 =0.5(3^31^1 - y 2 3 4 i ) y \2

C \4 = 0 . 5 ( x l 3 4 3 - * 21^ 2)^23 - 0.5x2iî3,31 + 0 -5 .vl3^ iy i2

Q 5 = 0 . 5 ( . r 2 i ^ i - - ' • • 32^ 3)^31 ~ 0 . 5 x 32^ 2 y i 2 + 0 . 5 . v 2 i <£2 >'23

C \ 6 = 0 . 5 ( . v 32^ 2 - - v 3 , ^ i ) y i 2 - 0 . 5 . r i 3 ^ 3 y 23 + 0 . 5 . r 32£ 3 3 / 3 i

C2i =0.5(yi2^2 - y 3i^3 ).r32 -0.5.v2i£iy3i + 0.5.Vi3£ iy i2

C 22 = 0 . 5 ( y 23£ 3 — 3 ; i 2 ^ l ) LVl 3 — 0 . 5 . r 32^ 2 y i 2 + 0 . 5 . v 2 i ^ 2 v 2 3

C 23 = 0 . 5 ( y 3 i ^ i — V 2 3 ^ 2 X v 2 1 ~ 0 . 5 . v i 3^ 3 >'23 + 0 . 5 . v 32£ 3 v 3 1

C 2 4 = 0 . 5 ( . v 13^ 3 — 1^2 )-r 3 2

C 25 = 0 . 5 ( . v 2 i ^ i — -V3 2 - = 3 )-v I 3

C 26 = 0 . 5 ( . v 3 i ^ 2 - . v 1 3 £ , > v 2 1


174

APPENDIX B

COORDINATE TRANSFORMATION

B .l Transformation of Lamina Stiffness Matrices

For a composite orthotropic lamina (see Figure B .l), the reduced lamina stiffness

matrix, [ Q ] , and shear stiffness matrix, [Q s], relate the lamina stress and strain in the

lamina material coordinate system (123 axes) as follows:

° 1 2 u 2t2O . = Q \2 0-22

.r I2 . 0 0

00

2 6 6 .£*2

V l 2

I 23

k l3 .244 0. 0 255 J

17'23

iri3(B.l)

These matrices are function of the material engineering constants:

2 n = 2i2 =^ 1 2 ^ - 2

2 2 2 —1 — 2 21 1 — 2 1 21 1 — ^ 12^21

Q . 6 6 ~ G l 2 ' Q 6 6 = (^ 2 3 ' Q&6 = ^ 13 ( B -2 )

The transformed stiffness and shear matrix for a general lamina with lamination angle 6

can be expressed in the laminate global coordinate axes (.rvz) by using the following

stress and strain transformation:

(B.3)&1 ' c 2 s 2 2 CS ' ° x 'O 2 » = s 2 c 2 - 2 CS O y • = [Ta } O y

.r 12 - c s c s 1 0 c 2 - s 2 Tn- . ^ XX

' c 2 S 2 CS ex£2 ■ = s 2 c 2 - c s £y ■ = IT£ } £y

Y \2 - 2 CS 2 CS c 2 - s 2 y * y . /■*>-.

[7 23

k i3 .

\y 23[ V l 3

cs

cs

- S 'c

- s c

\y v-S

* v z

y vcVxz J [Vxz

(B.4)

(B.5)

(B.6)

where C = c o s# S = s in # Substituting equations (B.3) through (B.6 ) in equation (B.l),

the lamina transformed reduced stiffness matrices are then:


175

= [ T a T l \ Q ] [ T £ ] (B.7)

011 012 016

012 022 026

.016 026 066.

044 045

.045 055.

B.2 Transformation of Piezoelectric Constants

The actuation strain induced by a piezoelectric layer expressed in material

B s h = [TS r l \ Q s ] i T S ] (B.8)

coordinates is:

£1 d i{■ = E r d i l

712 0(B.9)

where i = 3 for tradition monolithic piezoelectric actuators and i = 1 for MFC actuators.

Using the strain transformation matrix defined in equation (B.4), the piezoelectric

constant can then be express in the laminate reference coordinate as:

(B. 10)

d x d a

d y II d \ 2

d x y 0

Figure B. 1 Composite lamina with general fiber orientation


176

APPENDIX C

SOLUTION PROCEDURE FOR OPTIMAL OUTPUT FEEDBACK

Given the system state space matrices (A, B, C, D), performance index weighting

matrices (Q , R ), and initial conditions distribution (AT0), then the optimal output feedback

gain can be determined using the following iterative solution procedure:

1. Determine an initial stabilizing gain matrix, Ka, such that A-BKaC is stable.

2 . k-th iteration:

- Set A c = A-B{Ky)kC

- Solve for P k and S* matrices using (K y ) k and the following equations:

a J P + PAC + C T K y R K y C + Q = 0

A c S + s a J + X q = 0

- Set Jk = 0.5tr(PkX0)

If | J k - J k - i I < Tolerance, go to step 3, otherwise:

Determine the gain update direction:

AK y = R ~ 1B T P S C T (C S C T r 1 ~ ( K y )k

Update the gain using (K y )k + i = (K y ) k + a A K y

where a is a chosen constant

Repeat step 2

3. Terminate and set Ky= (Ky)k+i


177

CURRICULUM VITA

for

KHALED ABDEL-MOTAGALY

DEGREES:

• Doctor of Philosophy (Aerospace Engineering), Old Dominion University,

Norfolk, VA, Dec. 2001

• Master o f Science (Aerospace Engineering), Cairo University, Cairo, Egypt, July

1995

• Bachelor o f Science (Aerospace Engineering), Cairo University, Cairo, Egypt, July

1991

PROFESSIONAL CHRONOLOGY:

• Titan Systems Corp.- LinCom division, Houston, TX

Senior Systems Engineer, July 1999 - Present

• Eagle Aeronautics, Newport News, VA

Research Scientist, Jan. 1999 - June 1999

• Aerospace Engineering Dept., Old Dominion University, Norfolk, VA

Research and Teaching Assistant, Jan 1996 - Dec. 1998

• Aerospace Engineering Dept., University o f Cincinnati, Cincinnati, OH

Teaching Assistant, Sept. 1995 - Dec. 1995

• Aerospace Engineering Dept., Cairo University, Cairo, Egypt

Research and Teaching Assistant, Aug. 1991 - Aug. 1995

HONORS AND AWARDS:

• Member o f the national honor society Phi Kappa Phi

• Graduate Scholarship, Old Dominion University, Norfolk, VA, Jan. 1996 - Dec.

1998

• Graduate Scholarship, University of Cincinnati, Cincinnati, OH, Sept. 1995 - Dec.

1995

• Graduate Scholarship, Cairo University, Egypt, Dec. 1992 - Aug. 1995

• Egyptian Engineering Syndicate excellence award for the top aerospace

engineering student, 1991


178

• Honor degree, Cairo University, Egypt, 1991

• Student of the Year, Aerospace Engineering Department, Cairo University, Egypt,

1990-1991

SCHOLARLY ACTIVITIES COMPLETED:

R e f e r e e d J o u r n a l A r t i c l e s :

• Abdel-Motagaly, K., Duan, B., and Mei, C„ "Nonlinear response of composite

panels under combined acoustic excitation and aerodynamic pressure,” A I A A

J o u r n a l , Vol.38, No.9, Sept. 2000, pp. 1534-1542.

• Abdel-Motagaly, K„ Chen, R., and Mei, C., "Nonlinear flutter o f composite panels

under yawed supersonic flow using finite elements," A I A A J o u r n a l , Vol. 37, No. 9,

Sept. 1999, pp. 1025-1032.

• Mei, C., Abdel-Motagaly, K., and Chen, R., "Review of nonlinear panel flutter at

supersonic and hypersonic speeds,” A p p l i e d M e c h a n i c s R e v i e w , Vol. 52, No 10,

Oct. 1999, pp. 305-332.

• Hassan, S. D., El-Bayoumi, G. M., and Abdel-Motagaly, K., "Two new learning

algorithms for rules generation in self-organizing fuzzy logic controllers,” J o u r n a l

o f E n g i n e e r i n g a n d A p p l i e d S c i e n c e , Faculty of Engineering, Cairo University,

Vol. 43, No. 4, Aug. 1996, pp. 765-780.

• Hassan, S. D„ El-Bayoumi, G. M., and Abdel-Motagaly, K., "Longitudinal

autopilot design for a large scale flexible missile using LQR,” E n g i n e e r i n g

R e s e a r c h J o u r n a l , University of Helwan, Cairo, Egypt, Vol. 5, No. 6 , June 1995,

pp. 91-104.

S e l e c t e d R e f e r e e d P r o c e e d i n g s :

• Abdel-Motagaly, K„ Rombout, O., Gonzalez, R., Berrier, K., Hasan, D., Strack,

D„ and Rishikof, B., "Studies on the Attitude Control System Design for the

Crew Return Vehicle (X38),” A d v a n c e s i n t h e A s t r o n o m i c a l S c i e n c e s , G u i d a n c e

a n d C o n t r o l , Volume 104, 2000, pp. 281-297.

• Abdel-Motagaly, K., Duan, B., and Mei, C., “Nonlinear panel response at high

acoustic excitations and supersonic speeds,” P r o c e e d i n g s o f t h e 4 C fh A I A A

S t r u c t u r e , S t r u c t u r e D y n a m i c s , a n d M a t e r i a l s C o n f e r e n c e , St. Louis, MO, April

1999, pp. 1518-1529.


179

• Abdel-Motagaly, K., Chen, R., and Mei, C., “Effect o f flow angularity on

nonlinear supersonic flutter o f composite panels,” P r o c e e d i n g s o f t h e 4 ( f h A I A A

S t r u c t u r e , S t r u c t u r e D y n a m i c s , a n d M a t e r i a l s C o n f e r e n c e , St. Louis, MO, April

1999, pp. 1963-1972.

• Aledhaibi, A., Abdel-Motagaly, K., and Huang, J., ’’Design of self-organizing

fuzzy control for robot manipulator,” Proceedings of the 4 t h . J o i n t C o n f e r e n c e o n

I n f o r m a t i o n S c i e n c e s ( J C I S ) , Vol. I, Oct. 1998, Research Triangle Park, NC, pp.

73-75.

• Abdel-Motagaly, K. and Alberts, T., “Robust control design for a flexible

manipulator using Hoo/j.1 synthesis,” P r o c e e d i n g o f t h e A I A A G u i d a n c e ,

N a v i g a t i o n , a n d C o n t r o l C o n f e r e n c e , Boston, Massachusetts, Aug. 1998, pp. 365-

372.

• Abdel-Motagaly, K. and C. Mei, “On the control of nonlinear free vibration of a

simply supported beam,” P r o c e e d i n g s o f t h e 3 9 th A I A A S t r u c t u r e , S t r u c t u r e

D y n a m i c s , a n d M a t e r i a l s C o n f e r e n c e , Long Beach, CA, April 1998, pp. 3266-

3278.

• Lee, R. Y., Abdel-Motagaly, K., and Mei, C., “Fuzzy logic control for nonlinear

free vibration o f composite plates with piezoactuators,” P r o c e e d i n g s o f t h e 3 9 th

A I A A S t r u c t u r e , S t r u c t u r e D y n a m i c s , a n d M a t e r i a l s C o n f e r e n c e , Long Beach,

CA, April 1998, pp. 867-875.

• Hassan, S. D„ El-Bayoumi, G. M., and Abdel-Motagaly, K., “Application of

fuzzy logic control to the longitudinal autopilot o f a large scale flexible missile,”

3 r d I n t e r n a t i o n a l C o n f e r e n c e o n E n g i n e e r i n g M a t h e m a t i c s a n d P h y s i c s , Cairo,

Egypt, Dec. 1997, pp. 296-308.

• Hassan, S. D„ El-Bayoumi, G. M., and Abdel-Motagaly, K., “Attitude control of

large scale flexible missile using self-organizing fuzzy logic control,” 6 th

I n t e r n a t i o n a l C o n f e r e n c e in M e c h a n i c a l D e s i g n a n d P r o d u c t i o n , Cairo, Egypt,

Jan. 1996, pp.741-754.


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Finite Element Analysis and Active Control for Nonlinear